**2. Preliminaries**

#### **2.1 Control**

To enter into definition of what control is, we begin by explaining what a process is, which is referred to as a set of equipment or devices attached or implemented to perform operations that help fulfill a task [1]. To enforce this task, a series of additional devices are needed that regulate the process in general, which are called a control system as a whole [1].

Ogata [2] mentions four methods (belonging to the classical control or also called conventional control) for the design of a controller. These are the following:

• Analog

*Block diagram of a digital control system.*

*DOI: http://dx.doi.org/10.5772/intechopen.91817*

**Figure 2.**

• Digital

(k = 0,1,2, ..., n) [8].

**3. Parallel robots**

**117**

• Noise is a problem in all signal transmissions.

• It is easy to implement and modify digital controllers simply by changing

A digital or discrete time system is a dynamic system in which one or more

*Stewart-Gough Platform: Design and Construction with a Digital PID Controller Implementation*

The term parallel robot, also known as closed-chain robot or parallel manipulator, basically consists of a fixed base attached to an end effector or mobile base which can perform its movement based on the movement of the actuators that are located in such a way that they close the kinematic chain between both bases [9]. The principle of operation initially arose with James E. Gwinnett, who designed a platform for the entertainment industry in the 1928 patent application [10]. However, it was not until 1940 when W. Pollard designed the first industrial sparallel

After the first advances of the Stewart platforms applied to industrial processes, some companies began to make variants of these platforms, such as the case of the Redifon company that was asked by Lufthansa German Airlines to produce a simulator of flight, with the initial model designed for its then new Boeing 727 fleet. This simulator model included three axes, which gave the mechanism the mobility

variables may vary only at certain times called sampling indicated by kT

The block diagram of a digital control system is shown in **Figure 2** [9].

• They have the ability to reject noise.

robot for spray painting processes [11] (**Figures 3** and **4**).

the coefficient values.


In general, a control system needs a mathematical model that describes its behavior when receiving inputs [2]. For the Stewart platform, it can be determined by inverse kinematics [3].

The PID controller is the most common form of feedback, was an essential element of the first governors, and became a model or standard tool when process control emerged in the 1940s [4]. Consider a control loop of an input and an output. The block diagram [5] is shown in **Figure 1**.

Of the PID control family, there are members that involve the three actions: proportional (P), integral (I), and derivative (D) [5].

There are two control design techniques that are analog and digital, and for this they work with continuous time and discrete time systems, respectively. Many of the systems are described by differential equations, and analog control design techniques have become popular. Also most of the controllers are made of digital systems [6]. You can also discretize analog controllers to obtain digital controllers. Next, characteristics of control systems are mentioned, analog and digital [7]:

**Figure 1.** *SISO system block diagram.*

*Stewart-Gough Platform: Design and Construction with a Digital PID Controller Implementation DOI: http://dx.doi.org/10.5772/intechopen.91817*

• Analog

The final effector is the most interesting part of the robot, since its position and even orientation are the characteristics that determine if the robot is able to meet the precision necessary to implement and satisfy a need, whether industrial, educational, research, etc. It can have different applications, for example, orientation of satellites, flight simulators, and shakers (or also called agitators that are part of the

This document explains the design, construction, and implementation of a

To enter into definition of what control is, we begin by explaining what a process is, which is referred to as a set of equipment or devices attached or

implemented to perform operations that help fulfill a task [1]. To enforce this task, a series of additional devices are needed that regulate the process in general, which

Ogata [2] mentions four methods (belonging to the classical control or also called conventional control) for the design of a controller. These are the following:

In general, a control system needs a mathematical model that describes its behavior when receiving inputs [2]. For the Stewart platform, it can be determined

The PID controller is the most common form of feedback, was an essential element of the first governors, and became a model or standard tool when process control emerged in the 1940s [4]. Consider a control loop of an input and an output.

Of the PID control family, there are members that involve the three actions:

There are two control design techniques that are analog and digital, and for this they work with continuous time and discrete time systems, respectively. Many of the systems are described by differential equations, and analog control design techniques have become popular. Also most of the controllers are made of digital systems [6]. You can also discretize analog controllers to obtain digital controllers. Next, characteristics of control systems are mentioned, analog and digital [7]:

chemistry laboratory instruments), among others.

discrete PID control to a Stewart-Gough platform.

are called a control system as a whole [1].

**2. Preliminaries**

*Automation and Control*

• Root place

• Frequency response

• Modified PID controllers

The block diagram [5] is shown in **Figure 1**.

proportional (P), integral (I), and derivative (D) [5].

• PID controllers

by inverse kinematics [3].

**Figure 1.**

**116**

*SISO system block diagram.*

**2.1 Control**

	- They have the ability to reject noise.
	- It is easy to implement and modify digital controllers simply by changing the coefficient values.

A digital or discrete time system is a dynamic system in which one or more variables may vary only at certain times called sampling indicated by kT (k = 0,1,2, ..., n) [8].

The block diagram of a digital control system is shown in **Figure 2** [9].

### **3. Parallel robots**

The term parallel robot, also known as closed-chain robot or parallel manipulator, basically consists of a fixed base attached to an end effector or mobile base which can perform its movement based on the movement of the actuators that are located in such a way that they close the kinematic chain between both bases [9]. The principle of operation initially arose with James E. Gwinnett, who designed a platform for the entertainment industry in the 1928 patent application [10]. However, it was not until 1940 when W. Pollard designed the first industrial sparallel robot for spray painting processes [11] (**Figures 3** and **4**).

After the first advances of the Stewart platforms applied to industrial processes, some companies began to make variants of these platforms, such as the case of the Redifon company that was asked by Lufthansa German Airlines to produce a simulator of flight, with the initial model designed for its then new Boeing 727 fleet. This simulator model included three axes, which gave the mechanism the mobility

**Figure 3.** *James E. Gwinnett patent (industrial entertainment platform).*

needed to recreate the behavior of the aircraft. Redifon is currently in service since its commissioning in 1949, when it began with the production of flight simulators, trainers, and the development of new techniques [12] (**Figure 5**).

### **4. Inverse kinematics**

Inverse kinematics is a mathematical modeling of a manipulator, either serial (open kinematic chain) or parallel (closed kinematic chain). Such modeling requires as input parameters the position of the final effector in order to calculate the angles that exist between links and thus determine the position of each actuator in the reference *XYZ* coordinate space [13]. It should be noted that the final effector is the part of interest of any robot.

The fixed coordinates (*Fxyz*) are placed in the center of the fixed base, and the other mobile coordinate system (*Muvw*) is positioned in the center of the mobile platform (**Figure 6**).

Points *Fi* and *Mj* are the points of the joints between one actuator end with the fixed base and the other end of the actuator with the movable base, respectively. The separation angles between points *F*<sup>1</sup> and *F*2, *F*<sup>3</sup> and *F*4, and *F*<sup>5</sup> and *F*<sup>6</sup> are denoted by *θb*. Similarly, the angle of separation between points *M*<sup>1</sup> and *M*2, *M*<sup>3</sup> and *M*4, and *M*<sup>5</sup> and *M*<sup>6</sup> is denoted by *θp*. To locate the links or points *Fi*, use Eq. (1):

$$\begin{aligned} F\_i = \begin{bmatrix} F\_{Xi} \\ F\_Y \\ F\_{Zi} \end{bmatrix} = \begin{bmatrix} r\_b \cos\left(\mu\_i\right) \\ r\_b \text{sen}(\mu\_i) \\ \mathbf{0} \end{bmatrix} \end{aligned} \tag{1}$$

And for points *Mj*:

$$\mathbf{M}\_{j} = \begin{bmatrix} \mathbf{M}\_{Uj} \\ \mathbf{M}\_{\dot{V}} \\ \mathbf{M}\_{\dot{W}} \end{bmatrix} = \begin{bmatrix} r\_p \cos \left( \lambda\_j \right) \\ r\_p \text{sen} \left( \lambda\_j \right) \\ \mathbf{0} \end{bmatrix} \tag{2}$$

**Figure 5.**

**119**

**Figure 4.**

*First parallel robot designed by W. Pollard.*

*Stewart-Gough Platform: Design and Construction with a Digital PID Controller Implementation*

*DOI: http://dx.doi.org/10.5772/intechopen.91817*

*Flight simulator developed by Redifon for the Boeing 747 model.*

*Stewart-Gough Platform: Design and Construction with a Digital PID Controller Implementation DOI: http://dx.doi.org/10.5772/intechopen.91817*

**Figure 4.** *First parallel robot designed by W. Pollard.*

**Figure 5.** *Flight simulator developed by Redifon for the Boeing 747 model.*

needed to recreate the behavior of the aircraft. Redifon is currently in service since its commissioning in 1949, when it began with the production of flight simulators,

Inverse kinematics is a mathematical modeling of a manipulator, either serial

The fixed coordinates (*Fxyz*) are placed in the center of the fixed base, and the other mobile coordinate system (*Muvw*) is positioned in the center of the mobile

Points *Fi* and *Mj* are the points of the joints between one actuator end with the fixed base and the other end of the actuator with the movable base, respectively. The separation angles between points *F*<sup>1</sup> and *F*2, *F*<sup>3</sup> and *F*4, and *F*<sup>5</sup> and *F*<sup>6</sup> are denoted by *θb*. Similarly, the angle of separation between points *M*<sup>1</sup> and *M*2, *M*<sup>3</sup> and *M*4, and *M*<sup>5</sup> and *M*<sup>6</sup> is denoted by *θp*. To locate the links or points *Fi*, use Eq. (1):

> 2 6 4

> > 2 6 4

*rb* cos *μ<sup>i</sup>* ð Þ *rb*sen *μ<sup>i</sup>* ð Þ 0

> *rp* cos *λ <sup>j</sup>* � �

*rp*sen *λ <sup>j</sup>* � �

0

3 7

> 3 7

<sup>5</sup> (1)

<sup>5</sup> (2)

(open kinematic chain) or parallel (closed kinematic chain). Such modeling requires as input parameters the position of the final effector in order to calculate the angles that exist between links and thus determine the position of each actuator in the reference *XYZ* coordinate space [13]. It should be noted that the final effector

trainers, and the development of new techniques [12] (**Figure 5**).

*James E. Gwinnett patent (industrial entertainment platform).*

*Fi* ¼

*Mj* ¼

*FXi FY FZi*

*MUj MVj MWj*

2 6 4

2 6 4

**4. Inverse kinematics**

*Automation and Control*

**Figure 3.**

platform (**Figure 6**).

And for points *Mj*:

**118**

is the part of interest of any robot.

DH14. Obtain the transformation matrices *<sup>i</sup>*�<sup>1</sup>*Ai* defined as:

*cθ<sup>i</sup>* �*cαisθ<sup>i</sup> sαisθ<sup>i</sup> aicθ<sup>i</sup> sθ<sup>i</sup> cαicθ<sup>i</sup>* �*sαicθ<sup>i</sup> aisθ<sup>i</sup>* 0 *sα<sup>i</sup> cα<sup>i</sup> di* 00 0 1

*Stewart-Gough Platform: Design and Construction with a Digital PID Controller Implementation*

One of the main parameters to be defined within the control system implemen-

In general, the sampling period must be selected in compromise between a range of time that avoids the deterioration of the quality of the control that can produce a high value of T and the amount of calculations necessary to execute the control algorithm with small values that can produce information loss and frequency

In these, three cases are considered, which can be monitored based on the

• **Take the bandwidth of the system:** This considers the system as a closed loop system in which each of the elements and their respective frequency are raised. From this, the bandwidth of the system that will serve as a reference for the

• **Establishment time required for the transient response:** This method can be performed by simulation or experimental, since it considers obtaining a response time based on the reach of 63.3% of the final value in the transient response. In [16] an oscilloscope was used to measure the response curve and the Tao time by applying a pulse signal and a mechanism adapted to the actuators. In turn, a great advantage of this method is to observe the response curve and the actual efficiency of the actuators with respect to the data

• **Select the highest frequency component:** This is a method that allows an estimation of the sampling period based on the system component that requires sampling more frequently. Because currently the plant calculations

available elements, the ease of calculation, and the nature of the project.

tation is the sampling period T, which is a design element that allows different components to be selected among the most important ones such as the microcontroller. There are several methods for obtaining it; however, for this project the selection has been considered using the method described in [8] which selects the

period based on the commitment between the following factors:

(3)

*i*�1 *Ai* ¼

*DOI: http://dx.doi.org/10.5772/intechopen.91817*

1.The calculation time of the processor

3.Loss of information in the sampling

4.Response to disturbances

overlap (aliasing).

**121**

2.Numerical precision in the implementation

base frequency of sampling is determined.

provided by the manufacturer.

**6. Microcontroller selection**

**Figure 6.** *Isometric and top view of the platform with coordinate systems and actuator junction points.*
