**2.3. Analysis of the gearbox**

was developed by NI and contains a real-time processor which serves as a main control unit for the robot [10]. In addition, this platform includes a field-programmable gate array (FPGA) Xilinx Spartan-3 which is a reconfigurable device that executes programmed tasks in real time, that is, the active response of the system to external events. For this FPGA, a higher level of programming is possible using the NI LabVIEW® robotics software, which is a graphical

This mobile robot is programmed using an efficient algorithm to cover a trajectory that takes it to the box that needs palletizing. The aim is to illustrate the function of a box palletizer robot in industry. Yet, at this stage, it is only a prototype to show basic functions not involving

language. Programming languages like C, C++ or Java could also be used.

**Figure 1.** Global strategy for palletizer robot moves a box from place A to place B.

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heavy weights as those handled by an industrial robot.

The mechanical design of the robotic arm is based on a parallelogram arm (four bars) [11] and an arrangement of gear wheels (called gear train) [12] to transmit turning force and to provide a degree of freedom. The four-bar mechanism consists of two vertical bars of 8 cm in height and two horizontal bars of 6 cm in length. To implement our parallelogram arm, the four-bar mechanism is implemented twice, one for each servomotor used to move the arm up and down. Each motor is finally fixed to the gear train.

A 3D model in Solidworks® [13] of the different ratios of used gears is shown in **Figure 2**. Note that different ratios are used for drive transmission and for the moving arm in accordance with the desired speed ratios, which will be explained below.

The number of teeth on the gear is the most important parameter during gearbox design, because the speed of a final gear train only depends on this parameter, with 24 and 36 teeth being the most common sizes used. With *n* and *Z* being the desired angular velocity and the number of teeth of a gear, respectively, both parameters are directly related as:

$$n\_1 Z\_1 = n\_2 Z\_2 \tag{1}$$

where index *i* represents the motor (*i = 1*) and the driven (*i = 2*) gear. This equation provides the number of teeth required to provide a given angular velocity. If the rate *Z1 /Z2* is less than 1, the speed will be reduced. In our case, the gearbox uses eight gears, and we consider Z<sup>1</sup> = 8 teeth for gears 1, 3 and 5, and Z<sup>2</sup> = 24 for gears 2, 4 and 6, yielding a ratio of *1/3*. The last two gears 7 and 8 are considered as *Z1* = 16 and *Z2* = 36, respectively, with a ratio of 4/9. **Figure 3** shows the 3D design of the gear train implemented.

Another important parameter during gear train design is the relation between power supply and torque of the servomotor. The Lego servomotor datasheet establishes that

On the other hand, with *T1*

**Figure 3.** 3D design of the gearbox.

obtained by:

as the torque of the gearbox, then the power of the robotic arm is

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*P* = *T*<sup>1</sup> *n*<sup>1</sup> (2)

**Figure 2.** Ratios of the gearbox: three pairs of gears with a ratio of 1/3 are illustrated in (a), (b) and (c); in (d) the final pair of gears 7 and 8 has a ratio of 4/9.

for a power supply of 9 V (i.e., using 100%) the corresponding torque is 19 Ncm, and for 7.2 V (using only 75%) the torque is 16 Ncm [14]. For security reasons, we consider 15 Ncm as the maximal torque value provided by the robotic arm. In addition, **Table 1** shows experimental values of the angular velocity obtained at different values of power supply. Thus, considering a power supply of 75%, the angular velocity of the gearbox is around 95 rpm.

On the other hand, with *T1* as the torque of the gearbox, then the power of the robotic arm is obtained by:

$$\frac{\partial \mathbf{u}}{\partial \mathbf{u}} = \frac{\partial \mathbf{u}}{\partial \mathbf{u}} \cdot \nabla \mathbf{u} \mathbf{b}$$

$$P = \,^t T\_1 \,\text{u}\_1\tag{2}$$

**Figure 3.** 3D design of the gearbox.

for a power supply of 9 V (i.e., using 100%) the corresponding torque is 19 Ncm, and for 7.2 V (using only 75%) the torque is 16 Ncm [14]. For security reasons, we consider 15 Ncm as the maximal torque value provided by the robotic arm. In addition, **Table 1** shows experimental values of the angular velocity obtained at different values of power supply. Thus, considering a power supply of 75%, the angular velocity of the gearbox

**Figure 2.** Ratios of the gearbox: three pairs of gears with a ratio of 1/3 are illustrated in (a), (b) and (c); in (d) the final pair

is around 95 rpm.

of gears 7 and 8 has a ratio of 4/9.

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**Table 1.** Relation between power supply and torque of the servomotor (Lego datasheet [14]).

considering *T1 = 0.15* Nm and angular velocity of the motor as *n<sup>1</sup> = 9.9835* rad/s; thus, the driven power of the gearbox is *1.4975* W.

To obtain the internal velocities along the gear train, we use the torque equation defined as:

$$T\_z = T\_1 \cdot \frac{n\_1}{n\_2} \tag{3}$$

**3. Implementation of the palletizer robot**

four jaws to guarantee that object will be securely held.

**Figure 4.** 3D model of the robotic arm designed on Google SketchUP®.

in Section 2.3).

The robotic arm assembly requires bricks, girders, angle brackets, gearwheels, three servomotors and four touch sensors included in the Lego kit. Two of the servomotors move the mechanical part of the arm up and down, providing a degree of freedom. The third servomotor is used for closing and opening the griper. One of the touch sensors is at the base of the arm with the aim of sensing the lower position of the arm; similarly, a second touch sensor is located for sensing the higher position that can be reached by the arm. Third and fourth sensors are located on the gripper for measuring the opening and closing degrees controlled by the servomotor. A 3D model of the final robotic arm designed on Google SketchUP® [15] is illustrated on **Figure 4**, and the real robotic arm is shown in **Figure 5**; the gripper consists of

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The gearbox was designed in line with the required force and velocity to take and move an object from one place to another, without forcing the servomotors. The gearbox uses 8 gears: 3 of 16 teeth, 4 of 24 teeth and 1 of 36 teeth; it was designed following the analysis described

Once the robotic arm was built, we slightly modified the DaNI robot structure with the aim of mounting the robotic arm on it. In general, we replaced the rear omnidirectional wheel of

As the angular velocity *n<sup>1</sup>* is the same as the motor velocity, *n<sup>2</sup>* is given by:

$$n\_{\nu\_2} = n\_{\text{i}} \cdot \frac{Z\_{\text{i}}}{Z\_{\text{i}}} = 95.335 \cdot (8\% \text{a}) = 31.7783 \text{ rpm} \tag{4}$$

Then, the torque of gear 2 is obtained using Eq. (3), yielding

$$T\_2 = T\_1 \cdot \frac{n\_1}{n\_2} = 0.15 \cdot (^{\text{gs.333}}\text{<} \_{31.783}\text{>}) = 0.45 \text{ Nm} \tag{5}$$

Following this procedure, **Table 2** shows the torque values for 3–8 gears.

Therefore, torque and angular velocity at the output gear train are 9.1124 Nm and 1.5693 rpm, respectively. If the final torque is divided by the gravity force (g = 9.81 m/s2 ), then we obtain the mass in kg that the gripper can carry if the robotic arm length were 1 m. In our case, the robotic arm length is 0.20 m, a value that represents one-fifth of the reference value 1 m. By considering that, torque increases in the same factor as the orthogonal length to the applied force decreases, and the final mass that our robotic arm of 0.20 m in length can carry is as follows:

$$0.20\text{ m} \to 5(0.9289\text{ kg}) = 4.6445\text{ kg}\tag{6}$$


**Table 2.** Torque and angular velocity of gearbox.
