**2. System Definition**

Taking the component as a two degrees of freedom spherical mass whose position is fully defined by the horizontal and lateral linear position components of a point on it, i.e. point *P*, at any time with no orientation, it suffices to have a robot manipulator with two degrees of freedom to move the component to any point on the horizontal plane within its kinematic limits. The schematic view of the system involving the two-link robot manipulator and moving belt is submitted in **Figure 3** along with the corresponding definitions listed below.

*x* and *y*: horizontal and lateral axes of the inertial frame represented by *F0.*

*u* !ð Þ <sup>0</sup> <sup>1</sup> and *u* !ð Þ <sup>0</sup> <sup>2</sup> : unit vectors denoting the *x* and *y* axes of *F0.*

*O* and *A*: joints of the robot manipulator.

*a1* and *a2*: lengths of the first and second links of the robot manipulator.

*θ<sup>1</sup>* and *θ2*: relative rotation angles of the first and second joints of the robot manipulator.

*P*: point taken on the end effector of the robot manipulator.

*xP* and *yP*: horizontal and lateral position components of point *P.*

*S*: mid-point of the slot on the moving belt.

*Si*: changing points of the shape of the moving belt (*i* = 1, 2, 3, and 4).

*vS*: speed of the slot on the moving belt.

*xS* and *yS*: horizontal and lateral position components of point *S.*

*ρ*: turn radius of the moving belt.

*ψ*: rotation angles of the moving belt on its circular tip portions.

*L*: total length of the moving belt.

*d*: perpendicular distance between the connection point of the robot manipulator to the ground and the center line of the portion of the moving belt in the closest position to that point.

*g* !: gravity vector (*g* = 9.81 m/s2 ).

**Figure 3.** *System of the robot manipulator and moving belt.*

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

### **3. Robot Manipulator Kinematics**

SIMULINK® environment, it is decided that the present approach can be applied

Taking the component as a two degrees of freedom spherical mass whose position is fully defined by the horizontal and lateral linear position components of a point on it, i.e. point *P*, at any time with no orientation, it suffices to have a robot manipulator with two degrees of freedom to move the component to any point on the horizontal plane within its kinematic limits. The schematic view of the system involving the two-link robot manipulator and moving belt is submitted in **Figure 3**

*x* and *y*: horizontal and lateral axes of the inertial frame represented by *F0.*

<sup>2</sup> : unit vectors denoting the *x* and *y* axes of *F0.*

*P*: point taken on the end effector of the robot manipulator. *xP* and *yP*: horizontal and lateral position components of point *P.*

*a1* and *a2*: lengths of the first and second links of the robot manipulator. *θ<sup>1</sup>* and *θ2*: relative rotation angles of the first and second joints of the robot

*Si*: changing points of the shape of the moving belt (*i* = 1, 2, 3, and 4).

*d*: perpendicular distance between the connection point of the robot manipulator to the ground and the center line of the portion of the moving belt in the closest

*xS* and *yS*: horizontal and lateral position components of point *S.*

*ψ*: rotation angles of the moving belt on its circular tip portions.

).

on mounting lines to attain affordable and cheaper processes.

along with the corresponding definitions listed below.

*O* and *A*: joints of the robot manipulator.

*S*: mid-point of the slot on the moving belt.

*vS*: speed of the slot on the moving belt.

*ρ*: turn radius of the moving belt.

*L*: total length of the moving belt.

!: gravity vector (*g* = 9.81 m/s2

*System of the robot manipulator and moving belt.*

position to that point.

*g*

**Figure 3.**

**76**

**2. System Definition**

*Collaborative and Humanoid Robots*

!ð Þ <sup>0</sup>

*u* !ð Þ <sup>0</sup> <sup>1</sup> and *u*

manipulator.

In order to transform the guidance commands to the linear velocity components of the manipulator tip point into the angular speed variables of the joints, the kinematic relationships among those variables are considered.

Thus, the column vector of the position components of point *P* on the end effector (*rP*) can be written in terms of *θ<sup>1</sup>* and*θ <sup>2</sup>* in the next manner:

$$
\overline{r}\_P = a\_1 e^{j\theta\_1} + a\_2 e^{j\theta\_2} \tag{1}
$$

where *rP* ¼ *xP yP* � �*<sup>T</sup>* and *<sup>θ</sup>*<sup>12</sup> <sup>¼</sup> *<sup>θ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>θ</sup>*<sup>2</sup> with *<sup>j</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> while the letter "*e*" stands for the "exponential" operation.

Resolving Eq. (1) into its components, the equations given below come into the picture:

$$x\_P = a\_1 \cos\left(\theta\_1\right) + a\_2 \cos\left(\theta\_{12}\right) \tag{2}$$

$$y\_P = a\_1 \sin\left(\theta\_1\right) + a\_2 \sin\left(\theta\_{12}\right) \tag{3}$$

In velocity level, the following matrix expression is found by taking the time derivative of Eq. (1) with *θ* ¼ ½ � *θ*<sup>1</sup> *θ*<sup>2</sup> *T* :

$$
\dot{\overline{r}}\_P = \hat{f}\_P \dot{\overline{\theta}} \tag{4}
$$

In Eq. (4), the Jacobian matrix of the manipulator tip point is defined as follows:

$$\hat{f}\_P = \begin{bmatrix} -a\_1 \sin\left(\theta\_1\right) - a\_2 \sin\left(\theta\_{12}\right) & -a\_2 \sin\left(\theta\_{12}\right) \\ a\_1 \cos\left(\theta\_1\right) + a\_2 \cos\left(\theta\_{12}\right) & a\_2 \cos\left(\theta\_{12}\right) \end{bmatrix} \tag{5}$$

From Eq. (4), the angular velocities of the manipulator links can be obtained as given below:

$$
\dot{\overline{\theta}} = \hat{J}\_P^{-1} \dot{\overline{r}}\_P \tag{6}
$$

The "elbow-up" configuration of the manipulator in which the joint indicated by letter *A* in **Figure 3** becomes in the upper position is taken into account in the inverse kinematic calculation above.

Eventually, the linear acceleration equations come into the picture by taking the time derivative of Eq. (6):

$$
\ddot{\overline{\theta}} = \hat{J}\_P^{-1} \left( \ddot{\overline{r}}\_P - \dot{\hat{J}}\_P \dot{\overline{\theta}} \right) \tag{7}
$$

In Eq. (7), the time derivative of the tip point Jacobian matrix are determined in the forthcoming fashion with \_ *<sup>θ</sup>*<sup>12</sup> <sup>¼</sup> \_ *<sup>θ</sup>*<sup>1</sup> <sup>þ</sup> \_ *θ*2:

$$\dot{\hat{J}}\_P = \begin{bmatrix} -a\_1 \dot{\theta}\_1 \cos\left(\theta\_1\right) - a\_2 \dot{\theta}\_{12} \cos\left(\theta\_{12}\right) & -a\_2 \dot{\theta}\_{12} \cos\left(\theta\_{12}\right) \\\ -a\_1 \dot{\theta}\_1 \sin\left(\theta\_1\right) - a\_2 \dot{\theta}\_{12} \sin\left(\theta\_{12}\right) & -a\_2 \dot{\theta}\_{12} \sin\left(\theta\_{12}\right) \end{bmatrix} \tag{8}$$

#### **4. Dynamic Modeling of the Robot Manipulator**

The governing differential equations of motion of the robot manipulator schematized in **Figure 3** can be derived using the well-known virtual work method [9]. Neglecting the gravity vector ( *g* !) for the present engagement described on the horizontal plane, the equations of motion of the manipulator can be derived in the matrix form as follows:

$$
\overline{T} = \hat{M}(\overline{\theta})\,\overline{\overline{\theta}} + \hat{H}\left(\dot{\overline{\theta}}, \overline{\theta}\right)\dot{\overline{\theta}}\tag{9}
$$

other hand, it is easier and more practical to measure the joint speeds ( \_

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

the measured joint angles and their rates.

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

*θ*1*<sup>d</sup>* and \_

computed torque method [27–29]:

*<sup>M</sup>*^ ¼ �*m*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> \_

*<sup>θ</sup><sup>d</sup>* <sup>¼</sup> \_

*θ*1*<sup>d</sup>* \_ *θ*2*<sup>d</sup>* � �*<sup>T</sup>*

> *<sup>e</sup>* <sup>¼</sup> \_ *<sup>θ</sup><sup>d</sup>* � \_

*<sup>θ</sup><sup>d</sup>* <sup>þ</sup> *<sup>H</sup>*^ \_

*θ*, *θ*

based on the PI (proportional plus integral) control law [14, 30].

€*<sup>e</sup>* <sup>þ</sup> *<sup>M</sup>*^ �<sup>1</sup> \_

*θ*<sup>2</sup> sin ð Þ *θ*<sup>2</sup>

�, is obtained from Eq. (10) as follows:

<sup>2</sup> þ *Ic*<sup>2</sup> � � <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>2</sup>

0 2*ζ<sup>c</sup>*<sup>2</sup> *ω<sup>c</sup>*<sup>2</sup>

physical sense, *<sup>M</sup>*^ is invertible in all conditions and hence *<sup>M</sup>*^ �<sup>1</sup>

For a finite solution, the existence of *<sup>M</sup>*^ �<sup>1</sup>

<sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup>

width and damping parameters of the *i*

where *<sup>D</sup>*^ <sup>¼</sup> <sup>2</sup>*ζ<sup>c</sup>*1*ω<sup>c</sup>*<sup>1</sup> <sup>0</sup>

proportional and integral gain matrices, respectively, and *M*^ and *H*^ matrices are assumed to be accurately calculated. As can be seen, the proposed control system is

the error dynamics of the control system is obtained in the following manner:

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

> 2 1 1 0 � �.

matrix *M*^ . That is, the determinant of *M*^ must be nonzero. Here, the determinant of

<sup>1</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup>

For a second-order two-degree-of-freedom ideal system, the error dynamics can be defined using the forthcoming expression as *ωci* and *ζci* correspond to the band-

ters of the links of the manipulators disappears. Because this is not possible in

€*<sup>e</sup>* <sup>þ</sup> *<sup>D</sup>*^ \_

� � and *<sup>W</sup>*^ <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

<sup>2</sup> sin <sup>2</sup>

� never becomes zero unless any of the mass and inertia parame-

*<sup>c</sup>*<sup>1</sup> 0 0 *ω*<sup>2</sup> *c*2

" #

ð Þþ *θ*<sup>2</sup> *Ic*<sup>2</sup> � � <sup>þ</sup> *<sup>m</sup>*<sup>1</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup>

*th* link (*i* = 1 and 2), respectively [27]:

.

*<sup>e</sup>* <sup>þ</sup> *<sup>W</sup>*^ *<sup>e</sup>* <sup>¼</sup> <sup>0</sup> (18)

Inserting Eq. (15) into Eq. (9) and making the arrangements regarding Eq. (14),

*<sup>e</sup>* <sup>þ</sup> *<sup>M</sup>*^ �<sup>1</sup>

In order to make the steady state errors zero, the following control law including an integral action is designating upon the torque input of the manipulator as per the

*<sup>θ</sup>* <sup>þ</sup> *<sup>K</sup>*^ *<sup>p</sup> <sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*^*<sup>i</sup>*

desired and actual joint speeds (*e*) can be introduced as follows:

*<sup>T</sup>* <sup>¼</sup> *<sup>M</sup>*^ €

� � and *<sup>H</sup>*^ <sup>¼</sup> *<sup>H</sup>*^ \_

Introducing \_

with the column matrix \_

where *<sup>M</sup>*^ <sup>¼</sup> *<sup>M</sup>*^ *<sup>θ</sup>*

where \_

*M*^ , i.e. *M*^ � � �

> *M*^ � � �

**79**

� <sup>¼</sup> *Ic*<sup>1</sup> *<sup>m</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup>

As noticed, *M*^

� � �

than the linear velocity components of point P. For this reason, an indirect control scheme is designed in the present study such that the joint speeds are selected as control variables. In this situation, it is required to express the linear velocity components of point *P* in terms of the joint speeds. Here, the linear position and velocity components of point *P* can be calculated from Eqs. (2) through (4) using

*θ*2*<sup>d</sup>* to demonstrate the desired, or reference, joint speeds

ð

� � are defined. Also, *<sup>K</sup>*^ *<sup>p</sup>* and *<sup>K</sup>*^*<sup>i</sup>* stand for the

, the error column matrix between the

*θ* (14)

*e dt* (15)

*<sup>K</sup>*^*ie* <sup>¼</sup> <sup>0</sup> (16)

must be guaranteed by the invertible

exists.

1 *d*2 <sup>2</sup> (17)

*θ*<sup>1</sup> and \_ *θ*2)

In Eq. (9),*T* ¼ ½ � *T*<sup>1</sup> *T*<sup>2</sup> *<sup>T</sup>* stands for the torque column matrix with *T1* and *T2* which denote the control torques applied to the first and second joints of the robot manipulator, respectively. Here, the superscript "*T*" indicates the "transpose" operation. Also, the inertia and compound friction and Coriolis matrices [*M*^ *θ* and *<sup>H</sup>*^ \_ *θ*, *θ* ] are introduced in the following manner:

$$
\hat{M}(\overline{\theta}) = \begin{bmatrix} m\_{11} & m\_{12} \\ m\_{12} & m\_{22} \end{bmatrix} \tag{10}
$$

$$
\hat{H}\left(\dot{\overline{\theta}}, \overline{\theta}\right) = \begin{bmatrix} h\_{11} & h\_{12} \\ h\_{21} & h\_{22} \end{bmatrix} \tag{11}
$$

where *m1*, *m2*, *Ic1*, and *Ic2* correspond to the masses of the first and second links of the manipulator, and the moments of inertia of these links with respect to their mass centers indicated by *C1* and *C2*, respectively. Moreover, *b1* and *b2* are used to show the viscous friction coefficients at the first and second joints. With the definitions of *d*<sup>1</sup> ¼ j j *OC*<sup>1</sup> and *d*<sup>2</sup> ¼ j j *AC*<sup>2</sup> as additional length parameters, the following symbols are used in Eqs. (10) and (11).

$$\begin{aligned} m\_{11} &= m\_1 d\_1^2 + m\_2 \left[ a\_1^2 + d\_2^2 + 2a\_1 d\_2 \cos \left( \theta\_2 \right) \right] + I\_{c1} + I\_{c2}, m\_{12} \\ &= m\_2 d\_2 \left[ d\_2 + a\_1 \cos \left( \theta\_2 \right) \right] + I\_{c2}, m\_{22} = m\_2 d\_2^2 + I\_{c2}, h\_{11} \\ &= b\_1 - 2m\_2 a\_1 d\_2 \dot{\theta}\_2 \sin \left( \theta\_2 \right), h\_{12} = b\_2 - m\_2 a\_1 d\_2 \dot{\theta}\_2 \sin \left( \theta\_2 \right), h\_{21} \\ &= m\_2 a\_1 d\_2 \dot{\theta}\_1 \sin \left( \theta\_2 \right), \text{and } h\_{22} = b\_2. \end{aligned}$$

Substituting Eqs. (10) and (11) into Eq. (9), the following expressions are determined for *T1* and *T2*, respectively:

$$\begin{aligned} T\_1 &= \left[ m\_1 \dot{d}\_1^2 + m\_2 \left( a\_1^2 + d\_2^2 + 2a\_1 d\_2 \cos\left(\theta\_2\right) \right) + I\_{c1} + I\_{c2} \right] \ddot{\theta}\_1 \\ &+ \left[ m\_2 d\_2 \left( d\_2 + a\_1 \cos\left(\theta\_2\right) \right) + I\_{c2} \right] \ddot{\theta}\_2 + b\_1 \dot{\theta}\_1 + b\_2 \dot{\theta}\_2 \\ &- m\_2 a\_1 d\_2 \left( 2 \dot{\theta}\_1 \dot{\theta}\_2 + \dot{\theta}\_2^2 \right) \sin\left(\theta\_2 \right) \\ T\_2 &= \left[ m\_2 d\_2 \left( d\_2 + a\_1 \cos\left(\theta\_2\right) \right) + I\_{c2} \right] \ddot{\theta}\_1 + \left( m\_2 d\_2^2 + I\_{c2} \right) \ddot{\theta}\_2 \\ &+ b\_2 \dot{\theta}\_2 + m\_2 a\_1 d\_2 \dot{\theta}\_1^2 \sin\left(\theta\_2\right) \end{aligned} \tag{13}$$

### **5. Robot manipulator control system**

In order to keep the synchronization between the robot manipulator and moving belt during their engagement, it is more viable to make the control of the manipulator by considering its speed. That is, the components of the linear velocity vector of point *P* on the end effector of the manipulator in the horizontal plane become the parameters which should actually be controlled in a manner compatible with the command signals of the LHG law that are in the form of speed variables. On the

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

other hand, it is easier and more practical to measure the joint speeds ( \_ *θ*<sup>1</sup> and \_ *θ*2) than the linear velocity components of point P. For this reason, an indirect control scheme is designed in the present study such that the joint speeds are selected as control variables. In this situation, it is required to express the linear velocity components of point *P* in terms of the joint speeds. Here, the linear position and velocity components of point *P* can be calculated from Eqs. (2) through (4) using the measured joint angles and their rates.

Introducing \_ *θ*1*<sup>d</sup>* and \_ *θ*2*<sup>d</sup>* to demonstrate the desired, or reference, joint speeds with the column matrix \_ *<sup>θ</sup><sup>d</sup>* <sup>¼</sup> \_ *θ*1*<sup>d</sup>* \_ *θ*2*<sup>d</sup>* � �*<sup>T</sup>* , the error column matrix between the desired and actual joint speeds (*e*) can be introduced as follows:

$$
\overline{e} = \dot{\overline{\theta}}\_d - \dot{\overline{\theta}} \tag{14}
$$

In order to make the steady state errors zero, the following control law including an integral action is designating upon the torque input of the manipulator as per the computed torque method [27–29]:

$$\overline{T} = \hat{M}\ddot{\overline{\theta}}\_d + \hat{H}\dot{\overline{\theta}} + \hat{K}\_p\overline{e} + \hat{K}\_i \left[\overline{e}dt\right] \tag{15}$$

where *<sup>M</sup>*^ <sup>¼</sup> *<sup>M</sup>*^ *<sup>θ</sup>* � � and *<sup>H</sup>*^ <sup>¼</sup> *<sup>H</sup>*^ \_ *θ*, *θ* � � are defined. Also, *<sup>K</sup>*^ *<sup>p</sup>* and *<sup>K</sup>*^*<sup>i</sup>* stand for the proportional and integral gain matrices, respectively, and *M*^ and *H*^ matrices are assumed to be accurately calculated. As can be seen, the proposed control system is

based on the PI (proportional plus integral) control law [14, 30].

Inserting Eq. (15) into Eq. (9) and making the arrangements regarding Eq. (14), the error dynamics of the control system is obtained in the following manner:

$$
\ddot{\overline{\boldsymbol{e}}} + \hat{\boldsymbol{M}}^{-1} \left( \dot{\hat{\boldsymbol{M}}} + \hat{\boldsymbol{K}}\_{p} \right) \dot{\overline{\boldsymbol{e}}} + \hat{\boldsymbol{M}}^{-1} \hat{\boldsymbol{K}}\_{i} \overline{\boldsymbol{e}} = \overline{\mathbf{0}} \tag{16}
$$

$$
\dot{\hat{\boldsymbol{M}}} = -m\_{2} a\_{1} d\_{2} \dot{\theta}\_{2} \sin \left( \theta\_{2} \right) \begin{bmatrix} 2 & 1 \\ \mathbf{1} & \mathbf{0} \end{bmatrix} .\tag{17}
$$

For a finite solution, the existence of *<sup>M</sup>*^ �<sup>1</sup> must be guaranteed by the invertible matrix *M*^ . That is, the determinant of *M*^ must be nonzero. Here, the determinant of *M*^ , i.e. *M*^ � � � �, is obtained from Eq. (10) as follows:

$$\left|\dot{M}\right| = I\_{\varepsilon1} \left(m\_1 d\_1^2 + m\_2 d\_2^2 + I\_{\varepsilon2}\right) + m\_2 a\_1^2 \left[m\_2 d\_2^2 \sin^2(\theta\_2) + I\_{\varepsilon2}\right] + m\_1 m\_2 d\_1^2 d\_2^2 \tag{17}$$

As noticed, *M*^ � � � � never becomes zero unless any of the mass and inertia parameters of the links of the manipulators disappears. Because this is not possible in physical sense, *<sup>M</sup>*^ is invertible in all conditions and hence *<sup>M</sup>*^ �<sup>1</sup> exists.

For a second-order two-degree-of-freedom ideal system, the error dynamics can be defined using the forthcoming expression as *ωci* and *ζci* correspond to the bandwidth and damping parameters of the *i th* link (*i* = 1 and 2), respectively [27]:

$$
\ddot{\overline{e}} + \hat{D}\dot{\overline{e}} + \hat{W}\overline{e} = \overline{0} \tag{18}
$$

$$\text{where } \hat{D} = \begin{bmatrix} 2\zeta\_{c1}\alpha\_{c1} & 0\\ 0 & 2\zeta\_{c2}\alpha\_{c2} \end{bmatrix} \text{ and } \hat{W} = \begin{bmatrix} \alpha\_{c1}^2 & 0\\ 0 & \alpha\_{c2}^2 \end{bmatrix}.$$

where \_

Neglecting the gravity vector ( *g*

*Collaborative and Humanoid Robots*

In Eq. (9),*T* ¼ ½ � *T*<sup>1</sup> *T*<sup>2</sup>

symbols are used in Eqs. (10) and (11).

<sup>¼</sup> *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> \_

determined for *T1* and *T2*, respectively:

�*m*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> <sup>2</sup> \_

<sup>þ</sup>*b*<sup>2</sup> \_

**5. Robot manipulator control system**

<sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>2</sup>

*θ*<sup>1</sup> \_ *<sup>θ</sup>*<sup>2</sup> <sup>þ</sup> \_ *θ* 2 2

*<sup>θ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> \_

<sup>¼</sup> *<sup>b</sup>*<sup>1</sup> � <sup>2</sup>*m*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> \_

*<sup>m</sup>*<sup>11</sup> <sup>¼</sup> *<sup>m</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup>

*<sup>T</sup>*<sup>1</sup> <sup>¼</sup> *<sup>m</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup>

matrix form as follows:

*<sup>H</sup>*^ \_ *θ*, *θ* 

**78**

!) for the present engagement described on the

*<sup>T</sup>* stands for the torque column matrix with *T1* and *T2*

<sup>2</sup> þ *Ic*2, *h*<sup>11</sup>

*<sup>θ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> \_ *θ*2

> <sup>2</sup> þ *Ic*<sup>2</sup> €*θ*<sup>2</sup>

*θ*<sup>2</sup> sin ð Þ *θ*<sup>2</sup> , *h*<sup>21</sup>

*θ* (9)

and

(10)

(11)

(12)

(13)

horizontal plane, the equations of motion of the manipulator can be derived in the

which denote the control torques applied to the first and second joints of the robot manipulator, respectively. Here, the superscript "*T*" indicates the "transpose" operation. Also, the inertia and compound friction and Coriolis matrices [*M*^ *θ*

> <sup>¼</sup> *<sup>m</sup>*<sup>11</sup> *<sup>m</sup>*<sup>12</sup> *m*<sup>12</sup> *m*<sup>22</sup>

where *m1*, *m2*, *Ic1*, and *Ic2* correspond to the masses of the first and second links of the manipulator, and the moments of inertia of these links with respect to their mass centers indicated by *C1* and *C2*, respectively. Moreover, *b1* and *b2* are used to show the viscous friction coefficients at the first and second joints. With the definitions of *d*<sup>1</sup> ¼ j j *OC*<sup>1</sup> and *d*<sup>2</sup> ¼ j j *AC*<sup>2</sup> as additional length parameters, the following

> <sup>2</sup> þ 2*a*<sup>1</sup> *d*<sup>2</sup> cosð Þ *θ*<sup>2</sup> <sup>þ</sup> *Ic*<sup>1</sup> <sup>þ</sup> *Ic*2, *<sup>m</sup>*<sup>12</sup>

<sup>2</sup> þ 2*a*<sup>1</sup> *d*<sup>2</sup> cosð Þ *θ*<sup>2</sup> <sup>þ</sup> *Ic*<sup>1</sup> <sup>þ</sup> *Ic*<sup>2</sup> €*θ*<sup>1</sup>

sin ð Þ *θ*<sup>2</sup>

In order to keep the synchronization between the robot manipulator and moving belt during their engagement, it is more viable to make the control of the manipulator by considering its speed. That is, the components of the linear velocity vector of point *P* on the end effector of the manipulator in the horizontal plane become the parameters which should actually be controlled in a manner compatible with the command signals of the LHG law that are in the form of speed variables. On the

*<sup>θ</sup>*<sup>2</sup> sin ð Þ *<sup>θ</sup>*<sup>2</sup> , *<sup>h</sup>*<sup>12</sup> <sup>¼</sup> *<sup>b</sup>*<sup>2</sup> � *<sup>m</sup>*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *<sup>d</sup>*<sup>2</sup> \_

<sup>¼</sup> *<sup>h</sup>*<sup>11</sup> *<sup>h</sup>*<sup>12</sup> *h*<sup>21</sup> *h*<sup>22</sup> 

*<sup>θ</sup>* <sup>þ</sup> *<sup>H</sup>*^ \_

*θ*, *θ* \_

€

*<sup>T</sup>* <sup>¼</sup> *<sup>M</sup>*^ *<sup>θ</sup>*

] are introduced in the following manner:

*M*^ *θ*

*<sup>H</sup>*^ \_ *θ*, *θ* 

<sup>1</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup>

þ½ � *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup> <sup>ð</sup>*d*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup> cosð Þ *<sup>θ</sup>*<sup>2</sup> Þ þ *Ic*<sup>2</sup> €*θ*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>1</sup> \_

*<sup>T</sup>*<sup>2</sup> <sup>¼</sup> ½ � *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup> <sup>ð</sup>*d*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup> cosð Þ *<sup>θ</sup>*<sup>2</sup> Þ þ *Ic*<sup>2</sup> €*θ*<sup>1</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup>

*θ* 2 <sup>1</sup> sin ð Þ *θ*<sup>2</sup>

<sup>¼</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup> <sup>½</sup>*d*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup> cosð Þ *<sup>θ</sup>*<sup>2</sup> � þ *Ic*2, *<sup>m</sup>*<sup>22</sup> <sup>¼</sup> *<sup>m</sup>*<sup>2</sup> *<sup>d</sup>*<sup>2</sup>

*θ*<sup>1</sup> sin ð Þ *θ*<sup>2</sup> , and *h*<sup>22</sup> ¼ *b*2*:*

Substituting Eqs. (10) and (11) into Eq. (9), the following expressions are

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

$$
\hat{K}\_p = \hat{M}\hat{D} - \dot{\bar{M}} \tag{19}
$$

$$
\hat{K}\_i = \hat{M}\hat{W} \tag{20}
$$

*γ<sup>b</sup>* ¼

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

velocity vector of point *P* on the end effector (*v*

to be happen after a while as depicted in **Figure 5** with*v*

*γc*

\_

means of the next expression regarding Eqs. (6) and (27):

the linear velocity components of point *P* (\_

the velocity of point *S* and the ideal velocity of point *P* [26].

belt rollers, the position variables *xS*, *yS*, and *ψ* are obtained.

*rPd* <sup>¼</sup> *vP* cos *<sup>γ</sup><sup>c</sup>*

**8. Guidance law**

**Figure 4.**

derived as follows [4, 26]:

initiation of the engagement.

and \_

**81**

8 >>><

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

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

>>>:

*λ* ¼ *a* tan *yS* � *yP*

*π=*2 , *x*<sup>0</sup> ≤ *xS* <*x*<sup>1</sup> *ψ* � ð Þ *π=*2 , *x*<sup>1</sup> ≤*xS* <*x*<sup>2</sup> �*π=*2 , *x*<sup>2</sup> ≤ *xS* <*x*<sup>3</sup> *ψ* � ð Þ *π=*2 , *x*<sup>3</sup> ≤*xS* <*x*<sup>0</sup>

� �*=*ð Þ *xS* � *xP*

!

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

intercept point (*I*) at which the collision between the end effector and slot is going

In this law, in order for point *P* to catch point *S*, the guidance command (*γ<sup>c</sup>*

Here, using the measurements of *vS* by means of the appropriate sensors on the

In the application, the following column matrix including the reference values of

*m* � � sin *γ<sup>c</sup>*

In order to overcome the algebraic loop which occurs because the values of *vP*

The guidance commands can be expressed in terms of the angular speeds by

current component-picking motion of the manipulator is assigned to *vP* at the

\_ *<sup>θ</sup><sup>d</sup>* <sup>¼</sup> ^*<sup>J</sup>* �1 *P* \_

*rd* are dependent on each other, a nonzero value which is compatible with the

� � (25)

*Pactual*) towards the predicted

*m*:

*Pideal* which denote

*<sup>m</sup>*) is

! *<sup>S</sup>* and *v* !

*<sup>m</sup>* ¼ *λ* þ *a* sin ð Þ *vS=vP* sin *γ* ½ � ð Þ *<sup>b</sup>* � *λ* (26)

*rPd*) are formed using *γ<sup>c</sup>*

� � � � *<sup>T</sup>* (27)

*rPd* (28)

*m*

ð Þ rad (24)

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 or off-diagonal are updated at certain instants.
