# Algorithm to Generate Trajectories in a Robotic Arm Using an LCD Touch Screen to Help Physically Disabled People

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. LCD Touch Screen Operation

## 4. Algorithm Description

#### 4.1. Nomenclature

#### 4.2. Description of Conditions

#### 4.3. Linear Transformation

#### 4.4. Distribution and Trajectory Function

#### 4.5. Calculation of Velocity and Acceleration

## 5. Simulation and Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the four conditions and the two linear transformation. (

**a**) First linear transformation. (

**b**) Second linear transformation.

**Figure 2.**(

**a**) presents the trajectory ${S}_{\zeta a{t}_{s}}\left(x\right)$ for when $n=2$ and $n=12$ (

**b**) in order to obtain a different shape of the trajectory planning.

**Figure 5.**Different examples of trajectories for different values of $a$ and a total time equal to 5.

**Figure 6.**(

**a**) and (

**b**) show position, velocity, and acceleration and (

**c**) shows the Jerk of the trajectory in a second, starting in the position 0 and finalizing in 1 using different values of $a$ to change the velocity and acceleration (the parameter $a$ of (

**a**) is greater than the parameter $a$ of (

**b**)).

**Figure 7.**Two different results of trajectory to compare the final value of the trajectory, using three decimals and many decimals. On the right side of the figure, an approach is presented to observe the difference between the two final results obtained.

**Figure 8.**Illustration of the whole process to calculate the trajectory and the position-zero of the robotic arm where ${q}_{1},{q}_{2},{q}_{3}=0$ wich is presented on the right side and in the task space, the position is ${x}_{p}=1$, ${y}_{p}=2$ and $\alpha =0$, so it is had the coordinate (2, 0) with a direction equal to 0.

**Figure 9.**Guide of MATLAB, where the final position (rad), velocity (rad/s) and acceleration (rad/s

^{2}) values are shown, and the trajectory is seen with the joint position commands.

**Figure 10.**Example of a velocity and acceleration acquired of a linear-motion on each joint in the first condition with $a=0$. The velocity and acceleration are multiplied by $\pi $ to get the units (rad/s and rad/s

^{2}respectively). (

**a**) Velocity. (

**b**) Acceleration.

**Figure 11.**Example of a velocity and acceleration acquired of a curved-motion on each joint in the second condition using ${a}_{2},{a}_{3},{a}_{\alpha}0$. (

**a**) Velocity. (

**b**) Acceleration.

**Figure 12.**Example of a velocity and acceleration acquired on each joint in the third condition, using ${a}_{2}={a}_{3}={a}_{\alpha}<0$ to obtain a smoother linear-motion. (

**a**) Velocity. (

**b**) Acceleration.

**Figure 13.**Example of a velocity and acceleration acquired on each joint in the fourth condition using $-1.5\le {a}_{2}={a}_{3}={a}_{\alpha}\le 0$ to obtain a smooth not curved-motion. (

**a**) Velocity. (

**b**) Acceleration.

**Figure 14.**Simulation results of the alpha-join trajectory using distribution function proposed and cubic polynomial with a trajectory of 1 s.

**Table 1.**Results of the trajectory planning in the first condition, the coordinate $\left({x}_{p},{y}_{p}\right)$ is located in the first right triangle’s area.

$\mathsf{\alpha}.$ | $\left(\frac{{\mathit{y}}_{\mathit{p}}}{{\mathit{x}}_{\mathit{p}}}\right)$ | $\left({\mathit{\zeta}}_{2},{\mathit{\zeta}}_{3},{\mathit{\zeta}}_{\mathit{\alpha}}\right)$ | $\mathit{T}\left({\mathit{S}}_{2}\left(\mathit{x}\right),{\mathit{S}}_{3}\left(\mathit{x}\right),{\mathit{S}}_{\mathit{\alpha}}\left(\mathit{x}\right)\right)$ | $\dot{\mathit{S}}\left(\mathit{x}\right)$ | $\ddot{\mathit{S}}\left(\mathit{x}\right)$ |
---|---|---|---|---|---|

5.71 | $\frac{0.075}{0.75}$ | (2.534, 1.407, 5.978) | ($-0.345,0.75,0.031$) | (0.527, 1.152, 0.048) | (2.0371, 4.682, 0.187) |

11.3 | $\frac{0.17}{0.85}$ | (5.299, 5.666, 6.009) | ($-0.294,0.714,0.062$) | (0.077, 0.060, 0.047) | (0.3011, 0.2126, 0.177) |

21.8 | $\frac{0.415}{0.83}$ | (4.386, 6.521, 5.099) | ($-0.198,0.692,0.121$) | (0.146, 0.0333, 0.089) | (0.5669, 0.1299, 0.35) |

30.96 | $\frac{0.47}{0.78}$ | (6.510, 8.332, 5.296) | ($-0.176,0.699,0.172$) | (0.033, 0.009, 0.077) | (0.129, 0.041, 0.3005) |

41.98 | $\frac{0.83}{0.93}$ | (4.023, 3.972, 5.029) | ($-0.053,0.571,0.233$) | (0.188, 0.194, 0.093) | (0.7263, 0.7558, 0.361) |

45 | $\frac{1}{1}$ | (5.214, 4.803, 6.897) | ($-0.000,0.500,0.250$) | (0.082, 0.109, 0.025) | (0.313, 0.4192, 0.0991) |

**Table 2.**Results of the trajectory planning in the second condition, the coordinate $\left({x}_{p},{y}_{p}\right)$ is located in the second right triangle’s area and the three joints start with ${q}_{1}={q}_{2}={q}_{3}=0$ in the first result for when $\mathsf{\alpha}=45$ degrees.

$\mathsf{\alpha}$ | $\left(\frac{{\mathit{y}}_{\mathit{p}}}{{\mathit{x}}_{\mathit{p}}}\right)$ | $\left({\mathit{\zeta}}_{2},{\mathit{\zeta}}_{3},{\mathit{\zeta}}_{\mathit{\alpha}}\right)$ | $\mathit{T}\left({\mathit{S}}_{2}\left(\mathit{x}\right),{\mathit{S}}_{3}\left(\mathit{x}\right),{\mathit{S}}_{\mathit{\alpha}}\left(\mathit{x}\right)\right)$ | $\dot{\mathit{S}}\left(\mathit{x}\right)$ | $\ddot{\mathit{S}}\left(\mathit{x}\right)$ |
---|---|---|---|---|---|

45 | ($\frac{0.82}{0.82}$) | (2.162, 2.0277, 3.312) | ($-0.053,0.606,0.250$) | (0.0825, 0.1651, 0.399) | (0.5553, 1.110, 2.6193) |

63.434 | ($\frac{0.2}{0.4}$) | (2.162, 2.0227, 3.312) | ($-0.280,0.856,0.352$) | (0.3535, 0.389, 0.159) | (2.378, 2.619, 1.071) |

71.56 | ($\frac{0.93}{0.31}$) | (2.184, 2.4501, 4.463) | (0.060, 0.673, 0.397) | (0.579, 0.299, 0.0708) | (2.3352, 1.942, 0.4812) |

80.53 | ($\frac{0.77}{0.128}$) | (6.058, 3.816, 4.3267) | (0.075, 0.744, 0.447) | (0.023, 0.1109, 0.079) | (0.1596, 0.7536, 0.529) |

90 | $\left({x}_{p}\approx 0,{y}_{p}=1\right)$ | (3.680, 3.458, 4.2485) | (0.166, 0.666, 0.500) | (0.142, 0.1218, 0.082) | (0.9659, 0.8279, 0.558) |

**Table 3.**The trajectory planning results in the third condition are shown, the coordinate $\left({x}_{p},{y}_{p}\right)$ is located in the third right triangle’s area and the three joints start with ${q}_{1}={q}_{2}={q}_{3}=0$ in the first result for when $\mathsf{\alpha}=90$.

−α + 180 | $\left(\frac{{\mathit{y}}_{\mathit{p}}}{{\mathit{x}}_{\mathit{p}}}\right)$ | $\left({\mathit{\zeta}}_{2},{\mathit{\zeta}}_{3},{\mathit{\zeta}}_{\mathit{\alpha}}\right)$ | $\mathit{T}\left({\mathit{S}}_{2}\left(\mathit{x}\right),{\mathit{S}}_{3}\left(\mathit{x}\right),{\mathit{S}}_{\mathit{\alpha}}\left(\mathit{x}\right)\right)$ | $\dot{\mathit{S}}\left(\mathit{x}\right)$ | $\ddot{\mathit{S}}\left(\mathit{x}\right)$ |
---|---|---|---|---|---|

90 | $\left({x}_{p}\approx 0,{y}_{p}=0.61\right)$ | (3.271, 0.335, 1.000) | (0.103, 0.792, 0.500) | (0.161, 1.238, 0.781) | (0.568, 1.136, 4.511) |

100.8 | ($\frac{0.67}{-0.127}$) | (4.257, 6.143, 4.058) | (0.051, 0.778, 0.560) | (0.081, 0.022, 0.093) | (0.556, 0.150, 0.636) |

108.434 | ($\frac{0.56}{-0.186}$) | (4.111, 5.019, 4.559) | ($-0.006,0.809,0.602$) | (0.090, 0.048, 0.066) | (0.614, 0.327, 0.450) |

115.2 | ($\frac{0.69}{-0.32}$) | (7.097, 4.115, 4.733) | ($-0.014,0.751,0.640$) | (0.011, 0.090, 0.058) | (0.077, 0.612, 0.399) |

135 | ($\frac{0.97}{-0.97}$) | (7.753, 2.102, 3.184) | ($-0.009,0.518,0.750$) | (0.007, 0.363, 0.171) | (0.049, 2.471, 1.167) |

**Table 4.**Results of the trajectory planning in the fourth condition, the coordinate $\left({x}_{p},{y}_{p}\right)$ is located in the fourth right triangle’s area and the three joints start with ${q}_{1}={q}_{2}={q}_{3}=0$ in the first result for when $\mathsf{\alpha}=135$ and in the last result, $\mathsf{\alpha}=180$.

−α + 180 | $\left(\frac{{\mathit{y}}_{\mathit{p}}}{{\mathit{x}}_{\mathit{p}}}\right)$ | $\left({\mathit{\zeta}}_{2},{\mathit{\zeta}}_{3},{\mathit{\zeta}}_{\mathit{\alpha}}\right)$ | $\dot{\mathit{S}}\left(\mathit{x}\right)$ | $\ddot{\mathit{S}}\left(\mathit{x}\right)$ | |
---|---|---|---|---|---|

135 | ($\frac{0.81}{-0.81}$) | (4.442, 0.989, 0.695) | ($-0.055,0.611,0.750$) | (0.084, 0.927, 1.136) | (6.6471, 5.802, 7.121) |

141.34 | ($\frac{0.595}{-0.74}$) | (4.229, 4.184, 5.242) | ($-0.127,0.685,0.785$) | (0.106, 0.110, 0.053) | (1.193, 1.196, 1.172) |

153.43 | ($\frac{0.41}{-0.82}$) | (4.388, 6.821, 4.311) | ($-0.2008,0.696,0.852$) | (0.110, 0.017, 0.101) | (1.936, 1.173, 1.189) |

171 | ($\frac{0.101}{-0.64}$) | (3.422, 4.054, 5.273) | ($-0.345,0.790,0.950$) | (0.216, 0.139, 0.060) | (1.939, 1.930, 1.929) |

180 | ($\frac{0}{-0.19}$) | (4.005, 3.742, 5.321) | ($-0.4697,0.9394,1.00$) | (0.190, 0.228, 0.076) | (1.931, 1.943, 1.949) |

**Table 5.**Results obtained with the algorithm proposed in this work in comparison with the Cubic Polynomial Algorithm.

α | Trajectory Function-$\left(\mathit{\zeta}\mathit{a}{\mathit{t}}_{\mathit{s}}\right)$ | Cubic Polynomials |
---|---|---|

45 | 0.25(π) rad $\equiv $ 45$\xb0$ | 0.7633 radians $\equiv $ 43.73$\xb0$ |

71.56 | 0.3976(π) rad $\equiv $ 71.5624$\xb0$ | 1.2139 radians $\equiv $ 69.55$\xb0$ |

115.2 | 0.64(π) rad $\equiv $ 115.1988$\xb0$ | 1.9541 radians $\equiv $ 111.9656$\xb0$ |

180 | 1(π) rad $\equiv $ 180$\xb0$ | 3.053 radians $\equiv $ 174.941204$\xb0$ |

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Quiñonez, Y.; Mejía, J.; Zatarain, O.; Lizarraga, C.; Peraza, J.; Estrada, R.
Algorithm to Generate Trajectories in a Robotic Arm Using an LCD Touch Screen to Help Physically Disabled People. *Electronics* **2021**, *10*, 104.
https://doi.org/10.3390/electronics10020104

**AMA Style**

Quiñonez Y, Mejía J, Zatarain O, Lizarraga C, Peraza J, Estrada R.
Algorithm to Generate Trajectories in a Robotic Arm Using an LCD Touch Screen to Help Physically Disabled People. *Electronics*. 2021; 10(2):104.
https://doi.org/10.3390/electronics10020104

**Chicago/Turabian Style**

Quiñonez, Yadira, Jezreel Mejía, Oscar Zatarain, Carmen Lizarraga, Juan Peraza, and Rogelio Estrada.
2021. "Algorithm to Generate Trajectories in a Robotic Arm Using an LCD Touch Screen to Help Physically Disabled People" *Electronics* 10, no. 2: 104.
https://doi.org/10.3390/electronics10020104