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Research on joint torque reduction in robot manipulators has received considerable attention in recent years. Minimizing the computational complexity of torque optimization and the ability to calculate the magnitude of the joint torque accurately will result in a safe operation without overloading the joint actuators. This paper presents a mechanical design for a three dimensional planar redundant manipulator with the advantage of the reduction in the number of motors needed to control the joint angle, leading to a decrease in the weight of the manipulator. Many efforts have been focused on decreasing the weight of manipulators, such as using lightweight joints design or setting the actuators at the base of the manipulator and using tendons for the transmission of power to these joints. By using the design of this paper, only three motors are needed to control any n degrees of freedom in a three dimensional planar redundant manipulator instead of n motors. Therefore this design is very effective to decrease the weight of the manipulator as well as the number of motors needed to control the manipulator. In this paper, the torque of all the joints are calculated for the proposed manipulator (with three motors) and the conventional three dimensional planar manipulator (with one motor for each degree of freedom) to show the effectiveness of the proposed manipulator for decreasing the weight of the manipulator and minimizing driving joint torques.

Theoretically, for a structure of the robot manipulator one actuator can be mounted on each link to drive the next link via a speed reduction unit, but actuators and speed reducers installed on the distal end become the load for actuators installed on the proximal end of a manipulator, resulting in a bulky and heavy system [

Lightweight joint design based on a special rotary joint [

Provision of a powerful slider at the base to bear as much required driving force as possible [

The parallel mechanism is another method to reduce the mass and inertia of the manipulator [

Concentration of the actuators at the base and transmission of the power to each joint through tendons or a special transmission mechanism [

For a serial manipulator, direct kinematics are fairly straightforward, whereas inverse kinematics becomes very difficult. Reference [

It is mentioned earlier that the proposed manipulator could be used to reduce the weight of the manipulator which yields to a decrease in the size (power) of the motors used to control the manipulator. To show the effectiveness of the proposed manipulator in reducing the torques of its motors the inverse dynamic of the manipulator has been calculated mathematically. The inverse dynamic model provides the joint torques in terms of the joint positions, velocities and accelerations. For robot design, the inverse dynamic model is used to compute the actuator torques, which are needed to achive a desired motion [

The work presented in this paper is based on our previous work [

To control the motion of the end-effector of the manipulator shown of

Because the end-effector can follow any desired path by controlling three angles (_{1}_{2}_{3}_{1}_{2}_{3}_{3}

Elaborating further, the second motor is connected to the first link using a worm gear to control the angle _{2}

The third motor is connected to the second link using a worm gear for the same reasons it was used with the first link. Controlling the third motor means controlling the angle between the first link and the second link _{3}

The mechanism of the third link is shown in

To ensure that all the links move at the same joint angle, the ratio between the bevel gears of each planetary gear should be equal to one. This means the bevel gears of each planetary gear should have the same diameter and number of teeth. If this arm is fixed, we get:

And because the second gear will continue rotating with the same angular velocity, then:

Now the

For our manipulator it is desired to move both the arm and the second gear by the same angular velocity w which means:

To make the manipulator to have the ability to move in a three dimensional work space, a motor is added to the base of the manipulator to make the whole manipulator capable of rotating around the _{1}

To calculate the transformation matrix of the manipulator, the draft of the manipulator shown in _{1}_{2}_{5}_{1}

From the links parameters shown in _{i}_{i}_{i}_{i}

Finally we obtain the product of all six link transforms:

In this section, the torque of each joint is calculated. To show the effectiveness of the proposed manipulator, the joint torques are calculated using the proposed manipulator (using three motors only) and the conventional manipulators (a motor for each joint).

Let us assume for concreteness that the center of mass of each link is at its geometric center. For the manipulator used in our experiments, the mass of links without the motors are as follow: _{1}_{2}_{3}_{4}_{5}_{1}_{2}_{3}_{4}_{5}_{2}

The mass of each motor is 1,500 gm; for the manipulator of the proposed design, the first motor and the second motor are located on the base and not on the links themselves. Therefore, for our manipulator, the mass of the first link will be equal to the mass of this link (760 gm) plus the mass of the motor (1,500 gm) that controls the next links. Because there are no more motors, the mass of the links will be: _{1}_{2}_{3}_{4}_{5}

For the conventional three dimensional planar manipulator (one motor for each link), the mass of the first link will equal to the mass of link itself plus the mass of the motor which controls the second link position,

It is clearly noted how the proposed method could be used to decrease the weight of manipulator. Decreasing the weight leads to a decrease of the torques of each link. The next section shows the results of the torques of each joint when the end-effector is following a desired path. Using the Lagrangian formulation, the dynamical equations of motion of the manipulator is:

The first term in this equation is the inertia forces, the second term represents the Coriolis and centrifugal forces, and the third term gives the gravitational effects [

As shown by dynamics equations, increasing the weight of motors will increase the torques needed to control the manipulator. In order to decrease the effect of the motors weight on the inertia of manipulators, parallel manipulators are used, as we mentioned earlier. For example in reference [

This section shows the effectiveness of using the proposed manipulator to be used when it is desired to make the end-effector follow a desired path. This section has two examples. The first example calculates the torques using both manipulators (the proposed one and the conventional three dimensional planar manipulator) and shows how effective the proposed manipulator is in decreasing the torque of each joint required to move the manipulator. To verify the estimation results and compare between them and the results measured from the manipulator itself, the second example has been shown. This example shows the results if the torque using: (1) the conventional three dimensional planar manipulator with defined desired joint angles path, (2) the proposed manipulator with the defined desired joint angles path and finally (3) the proposed manipulator with the measured joint angles path when the joint angles follow the desired joint angles path.

Torque of each joint for both the manipulators is calculated to show the effectiveness of using the proposed manipulator to decrease the torque of each joint. Using the same manipulator with ^{T}_{2}

It should be remembered that when using the proposed manipulator, _{3}_{4}_{5}_{6}

First of all, it is noted that the torque of the sixth joint has the same value using both the manipulators because the sixth link has the same mass for both the manipulators, in other words the mass of the sixth link is equal to the mass of the link itself only because it does not hold any motor.

Secondly, as mentioned earlier for the proposed manipulator, the third motor should balance the torque of all the third, fourth, fifth and the sixth joint. In other words, the torque of the third motor should equal to (T_{3} + T_{4} + T_{5} + T_{6}) for the proposed manipulator.

The trajectory applied to robot in verification experiments in this case is:

_{3}_{4}_{5}_{6}

The results obtained from verification experiments indicate that there is a good agreement between the torque of the joint angles for the proposed manipulator using the estimated joint angles path (green) and the measured joint angles path (red). These figures show the effectiveness of the proposed manipulator in decreasing the torque of the joint angles using the proposed manipulator.

As mentioned in the first example that for the proposed manipulator, the third motor should balance the torque of all the third, fourth, fifth and the sixth joint, _{3} + T_{4} + T_{5} + T_{6}) for the proposed manipulator,

This paper presents a mechanical design for a three dimensional planar redundant manipulator. Theoretically, for each degree of freedom there should be one motor. However, in this design only three motors are needed to control any

This section explains the Dynamics of the manipulator used in experiments. In _{υi}_{wi}

_{υi}_{wi}

The link Jacobian submatrices, _{υi}

_{υ1}_{1}_{1}

To formulate the submatrices of _{wi}

Now to formulate the inertia matrices of the manipulator, assuming that the moving links are homogeneous with relatively small cross section, the inertia matrix ^{i}I_{i}

Now to find Ii inertia matrix of link i about its center of mass and expressed in the base link frame, the following equation is used:

Using

_{1}_{1}

Now to find the inertia manipulator matrix

Therefore,

The second term (the Coriolis and centrifugal forces term) and third term (gravitational effects term) of

(

The manipulator used in experiments [

The design of the second joint angle (first link with second motor) of the manipulator [

The design of the third joint angle (second link with third motor) of the manipulator [

The design of the fourth joint angle (third link) of the manipulator [

The last joint of the manipulator.

The mechanism of the first motor.

The manipulator used in experiments.

The position of mass for (

The values of the torques of the first joint using the both manipulators.

The values of the torques of the second joint using the both manipulators.

The values of the torques of the third joint using the both manipulators.

The values of the torques of the fourth joint using the both manipulators.

The values of the torques of the fifth joint using the both manipulators.

The values of the torques of the sixth joint using the both manipulators.

The values of the torques of the third motor using the both manipulators.

The values of the estimated and measured angular position, velocity and acceleration of the first joint angle (white: estimated, red: measured).

The values of the estimated and measured angular position, velocity and acceleration of the second joint angle (white: estimated, red: measured).

The values of the estimated and measured angular position, velocity and acceleration of the third joint angle (white: estimated, red: measured).

The torque of the first joint angle.

The torque of the second joint angle.

The torque of the third joint angle.

The torque of the fourth joint angle.

The torque of the fifth joint angle.

The torque of the sixth joint angle.

The torque of the third motor.

Link parameters of the manipulator.

i | α | a | d | θ |

1 | 90 | 0 | 0 | |

2 | 0 | |||

3 | 0 | 0 | ||

4 | 0 | 0 | ||

5 | 0 | 0 | ||

6 | 0 | 0 |

The mass of links for both the proposed and conventional manipulators.

| ||
---|---|---|

m2(gm) | 2,260 | 2,260 |

m3(gm) | 720 | 2,220 |

m4(gm) | 680 | 2,180 |

m5(gm) | 640 | 2,140 |

m6(gm) | 600 | 600 |