**6. Moving belt kinematics**

The horizontal and lateral position components of point *S* (*xS* and *yS*) showing the slot on the moving as shown in **Figure 3** are described at the changing points symbolized by *Si* (*i* = 1, 2, 3, and 4) by taking *vS* to be constant in the next manner:

$$\begin{aligned} \left(\mathbf{x}\_{\mathcal{S}}, \boldsymbol{y}\_{\mathcal{S}}\right) = \begin{cases} \begin{aligned} \boldsymbol{\varkappa}\_{0}, \boldsymbol{\jmath}\_{0} + \boldsymbol{\nu}\_{\boldsymbol{\varsigma}} (t - t\_{0}) & \text{, } \quad \boldsymbol{\varkappa}\_{0} \leq \boldsymbol{\varkappa}\_{\mathcal{S}} < \boldsymbol{\varkappa}\_{1} \\ \boldsymbol{\varkappa}\_{1} + \rho \left[ \cos\left(\boldsymbol{\wp}\right) + \mathbf{1} \right], \boldsymbol{\jmath}\_{1} + \rho \sin\left(\boldsymbol{\wp}\right) & \text{, } \quad \boldsymbol{\varkappa}\_{1} \leq \boldsymbol{\varkappa}\_{\mathcal{S}} < \boldsymbol{\varkappa}\_{2} \\ \boldsymbol{\varkappa}\_{2}, \boldsymbol{\jmath}\_{2} - \boldsymbol{\nu}\_{\boldsymbol{\varsigma}} (t - t\_{2}) & \text{, } \quad \boldsymbol{\varkappa}\_{2} \leq \boldsymbol{\varkappa}\_{\mathcal{S}} < \boldsymbol{\varkappa}\_{3} \\ \boldsymbol{\varkappa}\_{3} + \rho \left[ \cos\left(\boldsymbol{\wp}\right) - \mathbf{1} \right], \boldsymbol{\jmath}\_{3} + \rho \sin\left(\boldsymbol{\wp}\right) & \text{, } \quad \boldsymbol{\varkappa}\_{3} \leq \boldsymbol{\varkappa}\_{\mathcal{S}} < \boldsymbol{\varkappa}\_{0} \end{aligned} \end{aligned} \tag{21}$$

In Eq. (21), the position variables of point *S* at the *Si* location quantities are found by considering the following terms with the corresponding rotation angle at these points (*ψi*) (*π* = 3.14):

$$\begin{array}{l} \infty\_0 = d, \,\text{y}\_0 = \rho - (\text{L/2}), \,\text{y}\_0 = \pi \text{ rad}; \,\text{x}\_1 = d, \,\text{y}\_1 = (\text{L/2}) - \rho, \,\text{y}\_1 = \pi \text{ rad}; \,\text{x}\_2 \\\ = d + 2\rho, \,\text{y}\_2 = (\text{L/2}) - \rho, \,\text{y}\_2 = \mathbf{0}; \,\text{x}\_3 = d + 2\rho, \,\text{y}\_3 = \rho - (\text{L/2}), \,\text{and} \,\text{y}\_3 = \mathbf{0}. \end{array}$$

where *xi*, *yi*, and *ti* denote the horizontal and lateral position variables of point *S* at the *Si* location, and time parameter, respectively.

In Eq. (21),*ψ* can be determined from the following expression as a function of time (*t*) for *t*<sup>1</sup> ¼ *t*<sup>0</sup> þ *y*<sup>1</sup> � *y*<sup>0</sup> � � � �*=vS* � �, *<sup>t</sup>*<sup>2</sup> <sup>¼</sup> *<sup>t</sup>*<sup>1</sup> <sup>þ</sup> ð Þ *π ρ=vS* , and *<sup>t</sup>*<sup>3</sup> <sup>¼</sup> *<sup>t</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> � *<sup>y</sup>*<sup>3</sup> � � � �*=vS* � � with a specified t0 value:

$$\boldsymbol{\Psi} = \begin{cases} \boldsymbol{\pi} & , \quad \boldsymbol{\infty}\_{0} \le \boldsymbol{\pi}\_{i} < \boldsymbol{\pi}\_{1} \\ \boldsymbol{\pi} - \left[ \boldsymbol{v}\_{\mathcal{S}} \left( t - t\_{1} \right) / \rho \right] & , \quad \boldsymbol{\pi}\_{1} \le \boldsymbol{\pi}\_{i} < \boldsymbol{\pi}\_{2} \\ \boldsymbol{0} & , \quad \boldsymbol{\pi}\_{2} \le \boldsymbol{\pi}\_{i} < \boldsymbol{\pi}\_{3} \\ \boldsymbol{2}\boldsymbol{\pi} - \left[ \boldsymbol{v}\_{\mathcal{S}} \left( t - t\_{3} \right) / \rho \right] & , \quad \boldsymbol{\pi}\_{3} \le \boldsymbol{\pi}\_{i} < \boldsymbol{\pi}\_{0} \end{cases} \tag{22}$$

#### **7. Engagement geometry**

The engagement geometry between point *P* on the end effector of the robot manipulator and point *S* carried by the moving belt can be schematized on the horizontal plane as given in **Figure 4**.

Introducing *vP*, *γm*, *rS/P*, *γb*, and *λ* as the magnitude of the resulting velocity vector of point *P*, orientation angle of *vP* from the horizontal axis, relative position of point *S* with respect to point *P*, orientation angle of *vS* from the horizontal axis, and angle between *rS/P* and horizontal axis in **Figure 4**, respectively, *vP*, *γb*, and *λ* can be calculated using the equations below:

$$
v\_P = \sqrt{\dot{\mathbf{x}}\_P^2 + \dot{\mathbf{y}}\_P^2} \tag{23}
$$

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line DOI: http://dx.doi.org/10.5772/intechopen.97064*

#### **Figure 4.**

Finally, equating Eqs. (16) and (18) to each other, *K*^ *<sup>p</sup>* and *K*^*<sup>i</sup>* appear as follows:

In order to maintain the stability of the manipulator control systems throughout the engagement, the components of the matrices *K*^ *<sup>p</sup>* and *K*^*<sup>i</sup>* which may be diagonal

The horizontal and lateral position components of point *S* (*xS* and *yS*) showing the slot on the moving as shown in **Figure 3** are described at the changing points symbolized by *Si* (*i* = 1, 2, 3, and 4) by taking *vS* to be constant in the next manner:

*x*0, *y*<sup>0</sup> þ *vs*ð Þ *t* � *t*<sup>0</sup> , *x*<sup>0</sup> ≤*xS* < *x*<sup>1</sup>

*x*2, *y*<sup>2</sup> � *vs*ð Þ *t* � *t*<sup>2</sup> , *x*<sup>2</sup> ≤*xS* < *x*<sup>3</sup>

*x*<sup>1</sup> þ *ρ*½ � cosð Þþ *ψ* 1 , *y*<sup>1</sup> þ *ρ* sin ð Þ *ψ* , *x*<sup>1</sup> ≤*xS* <*x*<sup>2</sup>

*x*<sup>3</sup> þ *ρ*½ � cosð Þ� *ψ* 1 , *y*<sup>3</sup> þ *ρ* sin ð Þ *ψ* , *x*<sup>3</sup> ≤*xS* <*x*<sup>0</sup>

In Eq. (21), the position variables of point *S* at the *Si* location quantities are found by considering the following terms with the corresponding rotation angle at

*x0* ¼ *d*, *y0* ¼ *ρ* � ð Þ *L=*2 , *ψ<sup>0</sup>* ¼ *π* rad; *x1* ¼ *d*, *y1* ¼ ð Þ� *L=*2 *ρ*, *ψ<sup>1</sup>* ¼ *π* rad; *x2* ¼ *d* þ 2*ρ*, *y2* ¼ ð Þ� *L=*2 *ρ*, *ψ<sup>2</sup>* ¼ 0; *x3* ¼ *d* þ 2*ρ*, *y3* ¼ *ρ* � ð Þ *L=*2 , and *ψ<sup>3</sup>* ¼ 0*:*

where *xi*, *yi*, and *ti* denote the horizontal and lateral position variables of point *S*

In Eq. (21),*ψ* can be determined from the following expression as a function of

*π* � ½ � *vS* ð Þ *t* � *t*<sup>1</sup> *=ρ* , *x*<sup>1</sup> ≤*xs* <*x*<sup>2</sup>

2*π* � ½ � *vS* ð Þ *t* � *t*<sup>3</sup> *=ρ* , *x*<sup>3</sup> ≤*xs* <*x*<sup>0</sup>

The engagement geometry between point *P* on the end effector of the robot manipulator and point *S* carried by the moving belt can be schematized on the

Introducing *vP*, *γm*, *rS/P*, *γb*, and *λ* as the magnitude of the resulting velocity vector of point *P*, orientation angle of *vP* from the horizontal axis, relative position of point *S* with respect to point *P*, orientation angle of *vS* from the horizontal axis, and angle between *rS/P* and horizontal axis in **Figure 4**, respectively, *vP*, *γb*, and *λ*

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *x*\_ 2 *<sup>P</sup>* þ *y*\_ 2 *P*

*vP* ¼

� �, *<sup>t</sup>*<sup>2</sup> <sup>¼</sup> *<sup>t</sup>*<sup>1</sup> <sup>þ</sup> ð Þ *π ρ=vS* , and *<sup>t</sup>*<sup>3</sup> <sup>¼</sup> *<sup>t</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> � *<sup>y</sup>*<sup>3</sup>

*π* , *x*<sup>0</sup> ≤*xs* <*x*<sup>1</sup>

0 , *x*<sup>2</sup> ≤*xs* <*x*<sup>3</sup>

*M*^ (19)

(21)

�*=vS* � �

(23)

� � �

ð Þ rad (22)

*<sup>K</sup>*^*<sup>i</sup>* <sup>¼</sup> *<sup>M</sup>*^ *<sup>W</sup>*^ (20)

*<sup>K</sup>*^ *<sup>p</sup>* <sup>¼</sup> *<sup>M</sup>*^ *<sup>D</sup>*^ � \_

or off-diagonal are updated at certain instants.

**6. Moving belt kinematics**

*Collaborative and Humanoid Robots*

8 >>><

>>>:

at the *Si* location, and time parameter, respectively.

�*=vS*

� � �

8 >>><

>>>:

*ψ* ¼

horizontal plane as given in **Figure 4**.

can be calculated using the equations below:

*xS*, *yS* � � <sup>¼</sup>

these points (*ψi*) (*π* = 3.14):

time (*t*) for *t*<sup>1</sup> ¼ *t*<sup>0</sup> þ *y*<sup>1</sup> � *y*<sup>0</sup>

**7. Engagement geometry**

**80**

with a specified t0 value:

*Engagement geometry between the tip point of the manipulator and slot on the belt.*

$$\gamma\_b = \begin{cases} \pi/2 & , \quad \varkappa\_0 \le \varkappa\_S < \varkappa\_1 \\ \nu - (\pi/2) & , \quad \varkappa\_1 \le \varkappa\_S < \varkappa\_2 \\ \quad - \pi/2 & , \quad \varkappa\_2 \le \varkappa\_S < \varkappa\_3 \\ \nu - (\pi/2) & , \quad \varkappa\_3 \le \varkappa\_S < \varkappa\_0 \end{cases} \tag{24}$$

$$\lambda = \mathfrak{a} \tan \left[ (\mathcal{y}\_{\mathcal{S}} - \mathcal{y}\_{P}) / (\mathfrak{x}\_{\mathcal{S}} - \mathfrak{x}\_{P}) \right] \tag{25}$$

### **8. Guidance law**

In the LHG law, it is intended to keep the end effector of the manipulator always on the collision triangle that is formed by the end effector, slot, and predicted intercept point. For this purpose, the most convenient approach is to orient the velocity vector of point *P* on the end effector (*v* ! *Pactual*) towards the predicted intercept point (*I*) at which the collision between the end effector and slot is going to be happen after a while as depicted in **Figure 5** with*v* ! *<sup>S</sup>* and *v* ! *Pideal* which denote the velocity of point *S* and the ideal velocity of point *P* [26].

In this law, in order for point *P* to catch point *S*, the guidance command (*γ<sup>c</sup> <sup>m</sup>*) is derived as follows [4, 26]:

$$\gamma\_m^\varepsilon = \lambda + \mathfrak{a} \sin \left[ (\upsilon\_\mathcal{S}/\upsilon\_P) \sin \left( \chi\_b - \lambda \right) \right] \tag{26}$$

Here, using the measurements of *vS* by means of the appropriate sensors on the belt rollers, the position variables *xS*, *yS*, and *ψ* are obtained.

In the application, the following column matrix including the reference values of the linear velocity components of point *P* (\_ *rPd*) are formed using *γ<sup>c</sup> m*:

$$\dot{\vec{r}}\_{Pd} = \upsilon\_P \left[ \cos \left( \chi\_m^{\varepsilon} \right) \quad \sin \left( \chi\_m^{\varepsilon} \right) \right]^T \tag{27}$$

In order to overcome the algebraic loop which occurs because the values of *vP* and \_ *rd* are dependent on each other, a nonzero value which is compatible with the current component-picking motion of the manipulator is assigned to *vP* at the initiation of the engagement.

The guidance commands can be expressed in terms of the angular speeds by means of the next expression regarding Eqs. (6) and (27):

$$
\dot{\overline{\theta}}\_d = \hat{J}\_P^{-1} \dot{\overline{r}}\_{Pd} \tag{28}
$$

**Figure 5.** *Linear homing guidance law geometry.*

## **9. Computer simulations**

The numerical values considered in the computer simulations are submitted in **Table 1** along with the engagement block diagram in **Figure 6**.

The unit step responses at the first and second joints of the manipulator are submitted in **Figures 7** and **8** in which the discrete and continuous lines show the desired, or reference, and actual values of the joint angles, respectively. As shown, the desired joint speeds can be caught within the assigned bandwidth.


#### **Table 1.**

*Numerical values used in the computer simulations.*

In the designated engagement scenarios, it is assumed that the slot on the moving belt stands at point *S0* with *xS0* = 1.5 and *yS0* = 0.5 m at the instant when the robot manipulator is at rest. Furthermore, ramp-type angular speed inputs are applied to the manipulator joints in order for point *P* to attain its initial engagement velocity (*vPe*) at the end of 0.1 s. Here, the maximum angular speeds of the direct current (DC) electric motors connected to the joints are taken as 20 rad/s. The disturbance due to the nonlinear friction and noise on the sensors on the joints are

**Figure 7.**

**Figure 8.**

**83**

*Unit step response of the control system at the first joint.*

*DOI: http://dx.doi.org/10.5772/intechopen.97064*

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line*

*Unit step response of the control system at the second joint.*

**Figure 6.** *Block diagram for the robot manipulator-moving belt.*

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line DOI: http://dx.doi.org/10.5772/intechopen.97064*

**Figure 7.** *Unit step response of the control system at the first joint.*

**Figure 8.** *Unit step response of the control system at the second joint.*

In the designated engagement scenarios, it is assumed that the slot on the moving belt stands at point *S0* with *xS0* = 1.5 and *yS0* = 0.5 m at the instant when the robot manipulator is at rest. Furthermore, ramp-type angular speed inputs are applied to the manipulator joints in order for point *P* to attain its initial engagement velocity (*vPe*) at the end of 0.1 s. Here, the maximum angular speeds of the direct current (DC) electric motors connected to the joints are taken as 20 rad/s. The disturbance due to the nonlinear friction and noise on the sensors on the joints are

**9. Computer simulations**

*Linear homing guidance law geometry.*

*Collaborative and Humanoid Robots*

*Ic1* and *Ic2* 1.302 kg<sup>m</sup><sup>2</sup>

*Numerical values used in the computer simulations.*

*Block diagram for the robot manipulator-moving belt.*

**Table 1.**

**Figure 6.**

**82**

**Figure 5.**

The numerical values considered in the computer simulations are submitted in

The unit step responses at the first and second joints of the manipulator are submitted in **Figures 7** and **8** in which the discrete and continuous lines show the desired, or reference, and actual values of the joint angles, respectively. As shown,

**Parameter Value Parameter Value Parameter Value** *a1* and *a2* 1.25 m *b1* and *b2* 0.001 Nms/rad *L* 2 m *d1* and *d2* 0.625 m *ωc1* and *ωc2* 62.832 rad/s (=10 Hz) *ρ* 0.5 m *m1* and *m2* 10 kg *ζc1* and *ζc2* 0.707 *d* 1.5 m

**Table 1** along with the engagement block diagram in **Figure 6**.

the desired joint speeds can be caught within the assigned bandwidth.


**Table 2.**

*Simulation configurations considered.*

randomly varied in the ranges of 10 Nm and <sup>1</sup> <sup>10</sup><sup>3</sup> rad, respectively. The solver is selected to be the ODE5 (Dormand-Prince)-type solver with a fixed time step of 1 <sup>10</sup><sup>4</sup> s. The simulation configurations are designated as in **Table 2** along with the numerical values of the related parameters.

Having performed the computer simulations performed in the MATLAB® SIMULINK® environment, the results given in **Table 3** are attained. As samples, the engagement geometry for the configuration number 1 is given in **Figure 9** along with the plots for the changes of the velocity of point *P*, joint angles, joint speeds, and joint accelerations are submitted in **Figures 10**–**13**, respectively. Moreover, the engagement geometries for the sample configurations are plotted in **Figures 14** and **15**.

## **10. Discussion**

As given in **Figures 9, 14**, and **15** which belong to the designated simulation configurations at belt speeds from 0.5 to 2.5 m/s, it is observed that the tip point of the manipulator can catch the slot on the moving belt even at higher speeds. In the present work, the slot is caught by the manipulator near the left side of the belt. Actually, this placement strategy is desired in order to diminish the power consumption of the robot manipulator by keeping the motion distance short compared to the distance to the right side of the belt. Looking at the simulation data which are presented in the forms of relevant kinematic parameters of the manipulator in **Figures 10**–**13**, it can be verified that the angular speed values required at the joints of the manipulator can be attained even with industrial DC electric motors as well as the angular excursion demands.

#### **11. Conclusion**

Motion planning constitutes one of the significant issues in the development of autonomous system. In this context, guidance concept has been applied on munition developed to satisfy precise hitting requirements for recent years. Both theoretical studies and field implementations have revealed that the guidance algorithms have led the relevant munition to the desired target points successfully. Of course, the performance of the designated guidance scheme is directly related to the control systems whose primary function is to obey the commands generated by the

**Conf. No.**

**85**

1 2 3 4 5 **Table 3.** *Results attained from the considered*

 *simulation*

*configurations.*

0.299 0.308 0.267 0.327 0.745

31.339

30.082

30.654

29.050

25.000

**Engagement**

 **time (s) Maximum**

 **tip Point velocity (***vPmax***) (m/s)**

**Joint angles (°)**

> *θ<sup>1</sup>*

> > **Min.**

61.769

62.451

61.429

90.000

90.000

 180.000

164.369

46.956

20.146

 0.000

9.915

 20.091

 54.134

164.874

106.239

6.979

 20.189

17.313

 20.106

 50.059

161.766

105.679

6.268

 21.158

8.972

 20.602

 52.147

162.918

106.102

6.573

 20.288

8.469

 20.202

 44.093

159.876

104.421

6.127

 20.161

6.651

 20.085

 **Max.**

 **Min.**

 **Max.**

 **Min.**

 **Max.**

 **Min.**

 **Max.**

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line*

*θ2*

**Joint speeds (rad/s)**

*DOI: http://dx.doi.org/10.5772/intechopen.97064*

\_

\_

*θ***2**

*θ***<sup>1</sup>** *Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line DOI: http://dx.doi.org/10.5772/intechopen.97064*


**Table 3.**

*Results attained from the considered simulation configurations.*

randomly varied in the ranges of

with the numerical values of the related parameters.

the plots for the changes of the velocity of point

**point (m)**

*yP0*

0.5

*xP0*

*Collaborative and Humanoid Robots*

0.5

0.5

0.5

1.0

1.0

*Simulation configurations considered.*

accelerations are submitted in **Figures 10**

 10 

step of 1

**Conf. No.**

1

2

3

4

5

**Table 2.**

and **15** .

**10. Discussion**

**Figures 10**

**84**

**11. Conclusion**

the angular excursion demands.

10 N

0.5 5.0

0.5 5.0

ment geometries for the sample configurations are plotted in **Figures 14**

m and

**(m/s) Initial position of the tip**

solver is selected to be the ODE5 (Dormand-Prince)-type solver with a fixed time

Having performed the computer simulations performed in the MATLAB® SIMULINK® environment, the results given in **Table 3** are attained. As samples, the engagement geometry for the configuration number 1 is given in **Figure 9** along with

As given in **Figures 9, 14**, and **15** which belong to the designated simulation configurations at belt speeds from 0.5 to 2.5 m/s, it is observed that the tip point of the manipulator can catch the slot on the moving belt even at higher speeds. In the present work, the slot is caught by the manipulator near the left side of the belt. Actually, this placement strategy is desired in order to diminish the power consumption of the robot manipulator by keeping the motion distance short compared to the distance to the right side of the belt. Looking at the simulation data which are presented in the forms of relevant kinematic parameters of the manipulator in

of the manipulator can be attained even with industrial DC electric motors as well as

Motion planning constitutes one of the significant issues in the development of autonomous system. In this context, guidance concept has been applied on munition developed to satisfy precise hitting requirements for recent years. Both theoretical studies and field implementations have revealed that the guidance algorithms have led the relevant munition to the desired target points successfully. Of course, the performance of the designated guidance scheme is directly related to the control systems whose primary function is to obey the commands generated by the

–**13**, it can be verified that the angular speed values required at the joints

 1 10 

**Robot manipulator Moving belt**

*vPe***) (m/s)**

<sup>3</sup> 0.5

<sup>3</sup> 1.0

<sup>5</sup> 2.5

**Velocity at the beginning of the engagement (**

> 10

> 10

0.5 0.5 1.0

0.5 0.5 1.0

5 10 

<sup>4</sup> s. The simulation configurations are designated as in **Table 2** along

<sup>3</sup> rad, respectively. The

**velocity (** *v S* **)**

*P*, joint angles, joint speeds, and joint

–**13**, respectively. Moreover, the engage-

#### **Figure 9.**

*Engagement geometry for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

**Figure 10.**

*Change of the velocity of point P in time for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

similar approach in munition for the motion planning tasks of the manipulators. As a result of the present work, it can be deduced that the guidance-based approach leads to a successful placement for the components onto the intended slots in continuous engagement operations. This can be done even under considerable disturbing effects and undesirable changing speed conditions of the belt with lower

*Change of the joint speeds in time for the initial position components of the tip point of* xP0 *= 0.5 m and*

*Change of the joint angles in time for the initial position components of the tip point of* xP0 *= 0.5 m and*

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line*

power consumption levels.

yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

**Figure 11.**

**Figure 12.**

**87**

yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

*DOI: http://dx.doi.org/10.5772/intechopen.97064*

guidance law. For this purpose, several guidance and control approaches are proposed depending on the kind of the planned mission as can be encountered in the related literature.

Regarding the fact that robot manipulators are designed to achieve certain tasks which are usually specified before the execution, it can be a viable way to apply the *Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line DOI: http://dx.doi.org/10.5772/intechopen.97064*

**Figure 11.**

*Change of the joint angles in time for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

**Figure 12.**

*Change of the joint speeds in time for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

similar approach in munition for the motion planning tasks of the manipulators. As a result of the present work, it can be deduced that the guidance-based approach leads to a successful placement for the components onto the intended slots in continuous engagement operations. This can be done even under considerable disturbing effects and undesirable changing speed conditions of the belt with lower power consumption levels.

guidance law. For this purpose, several guidance and control approaches are proposed depending on the kind of the planned mission as can be encountered in the

*Change of the velocity of point P in time for the initial position components of the tip point of* xP0 *= 0.5 m and*

*Engagement geometry for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m*

Regarding the fact that robot manipulators are designed to achieve certain tasks which are usually specified before the execution, it can be a viable way to apply the

related literature.

yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

**Figure 10.**

**86**

**Figure 9.**

*with a moving belt velocity of 0.5 m/s.*

*Collaborative and Humanoid Robots*

**Figure 13.**

*Change of the joint accelerations in time for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

**Author details**

Mechanical Engineering Department, Gazi University, Ankara, Turkey

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Engagement geometry for the initial position components of the tip point of* xP0 *= 1.0 m and* yP0 *= 0.5 m*

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line*

*DOI: http://dx.doi.org/10.5772/intechopen.97064*

\*Address all correspondence to: bozkan37@gmail.com

provided the original work is properly cited.

Bülent Özkan

**89**

**Figure 15.**

*with a moving belt velocity of 2.5 m/s.*

#### **Figure 14.**

*Engagement geometry for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m with a moving belt velocity of 1.0 m/s.*

Although the applicability of the guidance and control approach on the robot manipulators is demonstrated by means of relevant computer simulations, there is not seen any serious difficulty to adapt the suggested concept into practice. This way, some of the robotic operations can be performed in an efficient manner.

*Guidance and Control of a Planar Robot Manipulator Used in an Assembly Line DOI: http://dx.doi.org/10.5772/intechopen.97064*

#### **Figure 15.**

*Engagement geometry for the initial position components of the tip point of* xP0 *= 1.0 m and* yP0 *= 0.5 m with a moving belt velocity of 2.5 m/s.*

## **Author details**

Bülent Özkan Mechanical Engineering Department, Gazi University, Ankara, Turkey

\*Address all correspondence to: bozkan37@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Although the applicability of the guidance and control approach on the robot manipulators is demonstrated by means of relevant computer simulations, there is not seen any serious difficulty to adapt the suggested concept into practice. This way, some of the robotic operations can be performed in an efficient manner.

*Engagement geometry for the initial position components of the tip point of* xP0 *= 0.5 m and* yP0 *= 0.5 m*

*Change of the joint accelerations in time for the initial position components of the tip point of* xP0 *= 0.5 m and*

**Figure 13.**

**Figure 14.**

**88**

*with a moving belt velocity of 1.0 m/s.*

yP0 *= 0.5 m with a moving belt velocity of 0.5 m/s.*

*Collaborative and Humanoid Robots*
