**Abstract**

In order to achieve higher productivity and lower cost requirements, robot manipulators have been enrolled in assembling processes in last decades as well as other implementation areas such as transportation, welding, mounting, and quality control. As a new application of this field, the control of the synchronous movements of a planar robot manipulator and moving belt is dealt with in this study. Here, the mentioned synchronization is tried to be maintained in accordance with a guidance law which leads the robot manipulator to put selected components onto the specific slots on the moving belt without interrupting the assembling process. In this scheme, the control of the manipulator is carried out by considering the PI (proportional plus integral) control law. Having performed the relevant computer simulations based on the engagement geometry between the robot manipulator and moving belt, it is verified that the mentioned pick-and-place task can be successfully accomplished under different operating conditions.

**Keywords:** Guidance and control, robot manipulator, linear homing guidance law, assembly line, automation

### **1. Introduction**

Robot manipulators have been utilized in many application areas since 1960's [1, 2]. In addition to their implementations in harsh and unusual environments involving tedious, hard, and hazardous tasks, the productivity, cost reduction, and time effectiveness considerations have put forward the use of the manipulators in the production and assembly applications [3–5].

Regarding the pick-and-place operations in which certain components are placed onto specific slots on a moving belt by means of the end effector of the robot manipulator that constitute the hand of the manipulator, the most common method is to make the placement of the component to the slot once they coincide. This attitude has been chosen by some famous vehicle manufacturers [6]. Since it is required to halt the moving belt at coincidences of the end effector of the robot and slot in this scheme, a discrete motion strategy is developed for this purpose. Even though this approach works well when relatively light components are under consideration, the increment in the component mass leads to higher acceleration requirements to speed up the belt right after the placement. In such a scheme with a robot manipulator, the belt should be halted at specific points in order to allow the

manipulator to put the component on the slot. In fact, this means using powerful actuators which are big and expensive in practice and hence it violates the cheapness demand.

guidance (PNG) law which is very popular in aerial systems is considered with different navigation constants under the absence and presence of obstacles on the collision trajectory between the manipulator and target object and relative com-

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

The control strategy is very significant for the realization of commands generated by the considered guidance law. For this purpose, certain control

norm-based robust control scheme supplemented by a Takagi-Sugeno type

methods are encountered in the literature for the robot manipulators such as H2/H<sup>∞</sup>

fuzzy control such that parameter uncertainties and nonlinear effects are accounted [22]. Similar to this work, the control of a prosthetic leg is handled regarding a stable robust adaptive impedance control [23]. The adaptive control of robot manipulators has become one of the most popular research areas for last decades. The control schemes based on classical laws such as PD (proportional plus derivative) law are proposed against parameter uncertainties and unmodeled disturbances. The effectiveness of the suggested approaches is then tried to be demonstrated by well-designed computer simulations and experimental studies

In this study, a guidance-based motion planning approach utilizing the linear homing guidance (LHG) law is proposed for the engagement problem between a planar two-link robot manipulator shown in **Figure 2** with an origin point O and moving belt in a continuous manner [20]. Although the LHG law generates the guidance commands in terms of the linear velocity components of the tip point of the manipulator, it is more reasonable to control the manipulator through the corresponding joint variables because the actuators are connected to the joints. Therefore, an indirect adaptive control system based on the computer torque method is designed by continuously updating the controller gains during the operation after transforming the guidance commands to the joint space [13, 26]. As per the data acquired from the computer simulations conducted in the MATLAB®

In order to synchronize the movements of the robot manipulator and moving belt in a continuous operation, it is a viable way to make them compatible in speed sense. This solution reduces to order of the robot manipulator dynamics two to one and thus most of the overshoots in the transient motion phase of the manipulator

puter simulations are performed [20].

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

can be prevented [21].

[24, 25].

**Figure 2.**

**75**

*Two-link robot manipulator [9].*

As a remedy to the preceding method, it seems reasonable to the motion of the belt even during placement. This results in diminishing the power need in operation and allows to use smaller and cheaper actuators, or motors [7]. On the other hand, it may not possible to coincide the end effector of the robot manipulator grasping the components and belt all the time due to uncertain factors such as nonlinear friction effects on the belt dynamics when larger and heavier parts are considered as in automotive industry. In order to compensate this weakness of conventional motion planning strategies based on making the placements upon the pre-calculated coincidence positions ignoring the probable uncertainties, "guidance" approach can be utilized in continuous-time engagements.

In addition to optimization-based motion planning schemes based on the minimum time and/or minimum energy expenditure criteria, a hybrid target point interception algorithm is proposed as schematized in **Figure 1** where the abbreviation AIPNG stands for the "augmented ideal proportional navigation guidance" for target catching [8–12]. In the mentioned studies in which the position information is often acquired by visual sensors, the engagement models including planar manipulators having two or three degrees of freedom, or two or three links, in general are validated through computer simulations [13, 14].

Guidance laws developed originally for the munitions against specific targets can be adapted to the motion planning of the robot manipulators that can be thought as "very short-range missiles" regarding their connections to the ground [15]. As an advantage over the munitions which have generally no thrust support during their guidance phase, the robot manipulators can be accelerated along their longitudinal axis [13]. In early applications, the robotic arms were tried to be guided by means of certain sensors placed on the end effectors such as optical sensors operating along with laser beams and visual sensors, i.e. cameras [16, 17]. As a distinguished implementation of guided robot manipulators, the guidance of micromanipulators utilized in microsurgery is accomplished by the visual guidance of the operator, i.e. surgeon [18]. The vision-based guidance approach is proposed for tele-robotic systems as well [2]. Moreover, the guidance of mobile robots is dealt with in swarm arrangements [19]. In another robotic application, the proportional navigation

**Figure 1.** *Hybrid target interception scheme [8].*

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

guidance (PNG) law which is very popular in aerial systems is considered with different navigation constants under the absence and presence of obstacles on the collision trajectory between the manipulator and target object and relative computer simulations are performed [20].

In order to synchronize the movements of the robot manipulator and moving belt in a continuous operation, it is a viable way to make them compatible in speed sense. This solution reduces to order of the robot manipulator dynamics two to one and thus most of the overshoots in the transient motion phase of the manipulator can be prevented [21].

The control strategy is very significant for the realization of commands generated by the considered guidance law. For this purpose, certain control methods are encountered in the literature for the robot manipulators such as H2/H<sup>∞</sup> norm-based robust control scheme supplemented by a Takagi-Sugeno type fuzzy control such that parameter uncertainties and nonlinear effects are accounted [22]. Similar to this work, the control of a prosthetic leg is handled regarding a stable robust adaptive impedance control [23]. The adaptive control of robot manipulators has become one of the most popular research areas for last decades. The control schemes based on classical laws such as PD (proportional plus derivative) law are proposed against parameter uncertainties and unmodeled disturbances. The effectiveness of the suggested approaches is then tried to be demonstrated by well-designed computer simulations and experimental studies [24, 25].

In this study, a guidance-based motion planning approach utilizing the linear homing guidance (LHG) law is proposed for the engagement problem between a planar two-link robot manipulator shown in **Figure 2** with an origin point O and moving belt in a continuous manner [20]. Although the LHG law generates the guidance commands in terms of the linear velocity components of the tip point of the manipulator, it is more reasonable to control the manipulator through the corresponding joint variables because the actuators are connected to the joints. Therefore, an indirect adaptive control system based on the computer torque method is designed by continuously updating the controller gains during the operation after transforming the guidance commands to the joint space [13, 26]. As per the data acquired from the computer simulations conducted in the MATLAB®

**Figure 2.** *Two-link robot manipulator [9].*

manipulator to put the component on the slot. In fact, this means using powerful actuators which are big and expensive in practice and hence it violates the cheap-

As a remedy to the preceding method, it seems reasonable to the motion of the belt even during placement. This results in diminishing the power need in operation and allows to use smaller and cheaper actuators, or motors [7]. On the other hand, it may not possible to coincide the end effector of the robot manipulator grasping the components and belt all the time due to uncertain factors such as nonlinear friction effects on the belt dynamics when larger and heavier parts are considered as in automotive industry. In order to compensate this weakness of conventional motion planning strategies based on making the placements upon the pre-calculated coincidence positions ignoring the probable uncertainties, "guidance" approach can be

In addition to optimization-based motion planning schemes based on the mini-

Guidance laws developed originally for the munitions against specific targets can be adapted to the motion planning of the robot manipulators that can be thought as "very short-range missiles" regarding their connections to the ground [15]. As an advantage over the munitions which have generally no thrust support during their guidance phase, the robot manipulators can be accelerated along their longitudinal axis [13]. In early applications, the robotic arms were tried to be guided by means of certain sensors placed on the end effectors such as optical sensors operating along with laser beams and visual sensors, i.e. cameras [16, 17]. As a distinguished implementation of guided robot manipulators, the guidance of micromanipulators utilized in microsurgery is accomplished by the visual guidance of the operator, i.e. surgeon [18]. The vision-based guidance approach is proposed for tele-robotic systems as well [2]. Moreover, the guidance of mobile robots is dealt with in swarm arrangements [19]. In another robotic application, the proportional navigation

mum time and/or minimum energy expenditure criteria, a hybrid target point interception algorithm is proposed as schematized in **Figure 1** where the abbreviation AIPNG stands for the "augmented ideal proportional navigation guidance" for target catching [8–12]. In the mentioned studies in which the position information is often acquired by visual sensors, the engagement models including planar manipulators having two or three degrees of freedom, or two or three links, in

general are validated through computer simulations [13, 14].

ness demand.

*Collaborative and Humanoid Robots*

**Figure 1.**

**74**

*Hybrid target interception scheme [8].*

utilized in continuous-time engagements.

SIMULINK® environment, it is decided that the present approach can be applied on mounting lines to attain affordable and cheaper processes.

**3. Robot Manipulator Kinematics**

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

� �*<sup>T</sup>*

derivative of Eq. (1) with *θ* ¼ ½ � *θ*<sup>1</sup> *θ*<sup>2</sup>

inverse kinematic calculation above.

time derivative of Eq. (6):

the forthcoming fashion with \_

^*JP* <sup>¼</sup> �*a*<sup>1</sup> \_

�*a*<sup>1</sup> \_

\_

**77**

for the "exponential" operation.

where *rP* ¼ *xP yP*

picture:

given below:

In order to transform the guidance commands to the linear velocity components

*rP* <sup>¼</sup> *<sup>a</sup>*<sup>1</sup> *<sup>e</sup> <sup>j</sup> <sup>θ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup> *<sup>e</sup> <sup>j</sup> <sup>θ</sup>*<sup>12</sup> (1)

*xP* ¼ *a*<sup>1</sup> cosð Þþ *θ*<sup>1</sup> *a*<sup>2</sup> cosð Þ *θ*<sup>12</sup> (2) *yP* ¼ *a*<sup>1</sup> sin ð Þþ *θ*<sup>1</sup> *a*<sup>2</sup> sin ð Þ *θ*<sup>12</sup> (3)

*<sup>a</sup>*<sup>1</sup> cosð Þþ *<sup>θ</sup>*<sup>1</sup> *<sup>a</sup>*<sup>2</sup> cosð Þ *<sup>θ</sup>*<sup>12</sup> *<sup>a</sup>*<sup>2</sup> cosð Þ *<sup>θ</sup>*<sup>12</sup> � � (5)

�<sup>1</sup> <sup>p</sup> while the letter "*e*" stands

*θ* (4)

*rP* (6)

� � (7)

*θ*<sup>12</sup> cosð Þ *θ*<sup>12</sup>

of the manipulator tip point into the angular speed variables of the joints, the

Thus, the column vector of the position components of point *P* on the end

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

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

In velocity level, the following matrix expression is found by taking the time

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

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

The "elbow-up" configuration of the manipulator in which the joint indicated by

Eventually, the linear acceleration equations come into the picture by taking the

In Eq. (7), the time derivative of the tip point Jacobian matrix are determined in

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].

^*JP* \_ *θ*

*<sup>θ</sup>*<sup>12</sup> cosð Þ� *<sup>θ</sup>*<sup>12</sup> *<sup>a</sup>*<sup>2</sup> \_

*<sup>θ</sup>*<sup>12</sup> sin ðÞ� *<sup>θ</sup>*<sup>12</sup> *<sup>a</sup>*<sup>2</sup> \_ *<sup>θ</sup>*<sup>12</sup> sin ð Þ *<sup>θ</sup>*<sup>12</sup> " # (8)

^*JP* <sup>¼</sup> �*a*<sup>1</sup> sin ð Þ� *<sup>θ</sup>*<sup>1</sup> *<sup>a</sup>*<sup>2</sup> sin ð Þ� *<sup>θ</sup>*<sup>12</sup> *<sup>a</sup>*<sup>2</sup> sin ð Þ *<sup>θ</sup>*<sup>12</sup>

*T* :

\_ *rP* <sup>¼</sup> ^*JP* \_

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

letter *A* in **Figure 3** becomes in the upper position is taken into account in the

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

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

*<sup>θ</sup>*<sup>12</sup> <sup>¼</sup> \_

*<sup>θ</sup>*<sup>1</sup> cosð Þ� *<sup>θ</sup>*<sup>1</sup> *<sup>a</sup>*<sup>2</sup> \_

*<sup>θ</sup>*<sup>1</sup> sin ð Þ� *<sup>θ</sup>*<sup>1</sup> *<sup>a</sup>*<sup>2</sup> \_

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

kinematic relationships among those variables are considered.

effector (*rP*) can be written in terms of *θ<sup>1</sup>* and*θ <sup>2</sup>* in the next manner:

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