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Sensors 2012 12 6869 6892 doi 10 3390 s120606869 sensors ISSN 1424 8220 Article Joint Torque Reduction of a Three Dimensional Redundant Planar Manipulator Samer Yahya 1 Mahmoud Moghavvemi 1 2 and Haider Abbas F Almurib 3 1 Center of Research in Applied Electronics CRAE University of Malaya Kuala Lumpur 50603 Malaysia E Mail mahmoud um edu my 2 Faculty of Electrical and Computer Engineering University of Tehran P O Box 14399 57131 Tehran Iran 3 Department of Electrical E Mail haider abbas nottingham edu my Author to whom correspondence should be addressed E Mail smryahya Tel 60 172 841 560 Received 1 April 2012 in revised form 2 May 2012 Accepted 22 May 2012 Published 25 May 2012 Abstract 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 OPEN ACCESS Sensors 2012 12 6870 Keywords redundant manipulator dynamics robot rotary encoders joint torques reduction 1 Introduction 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 1 To reduce the weight and the inertia of a robot manipulator many mechanisms have been proposed so far to remove the weight restriction Some reported by 2 3 include a Lightweight joint design based on a special rotary joint 4 6 b Provision of a powerful slider at the base to bear as much required driving force as possible 7 c The parallel mechanism is another method to reduce the mass and inertia of the manipulator 8 A typical parallel manipulator consists of a moving platform that is connected with a fixed base by several limbs Generally the number of degrees of freedom of a parallel manipulator is equal to the number of its limbs The actuators are usually mounted on or near the base which contributes to reduce the inertia of manipulators and d Concentration of the actuators at the base and transmission of the power to each joint through tendons or a special transmission mechanism 2 3 9 This mechanism allows the actuators to be situated remotely on the manipulator base allowing the manipulator to be made more lightweight and compact For a serial manipulator direct kinematics are fairly straightforward whereas inverse kinematics becomes very difficult Reference 10 proposes a fused smart sensor network to estimate the forward kinematics of an industrial robot while reference 11 measures the range data with respect to the robot base frame using the robot forward kinematics and the optical triangulation principle The inverse kinematics problem is much more interesting and its solution is more useful but one of the difficulties of inverse kinematics is that when a manipulator is redundant it is anticipated that the inverse kinematics has an infinite number of solutions This implies that for a given location of the manipulator s end effector it is possible to induce a self motion of the structure without changing the location of the end effector In this paper we depend on our prior works 12 13 which present a new method to solve the problem of multi solutions of a three dimensinal planar redundant manipulator Because this paper explains the dynamic of the manipulator and not its kinematics the inverse kinematics methods will not be explained here For more details about the inverse kinematics of redundant manipulators our works 14 16 can be checked 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 17 Several approaches have been proposed to model the dynamics of robots The most Sensors 2012 12 6871 frequently employed in robotics are the Lagrange formulation and the Newton Euler formulation Because the Lagrange formulation is conceptually simple and systematic 18 it has been used in this paper The Lagrange formulation provides a description of the relationship between the joint actuator forces and the motion of the mechanism and fundamentally operates on the kinetic and potential energy in the system 19 The work presented in this paper is based on our previous work 14 which presents a mechanical design for a three dimensional planar redundant manipulator which guarantees to decrease the weight of the manipulator by decreasing the number of motors needed to control it Because the inverse kinematics model gives an infinite number of solutions for a redundant manipulator consequently secondary performance criteria can be optimized 17 such as avoiding singular configurations and minimizing driving joint torques Reference 14 studied the kinematics of the manipulator of this paper and showed in details its ability to avoid singular configurations A comparison of the manipulability index values and the manipulability ellipsoids for the manipulator is made with the manipulability index values and the manipulability ellipsoids of the PUMA arm to show the effectiveness of using the proposed manipulator to avoid singularity In this paper the dynamics of this manipulator are explained in detail The contribution of this work is to explain the ability of this manipulator for joint torque minimization The links and motors mass distribution is studied for both the proposed with three motors and conventional manipulators six motors The driving joint torques have been studied for the proposed manipulator for each joint and the results are compared with the results of the conventional manipulators to show the effectiveness of this manipulator for minimizing driving joint torques 2 The Mechanical Design of the Manipulator To control the motion of the end effector of the manipulator shown of Figure 1 a all the motors of the manipulator should be controlled For example to control a five links planar redundant manipulator with the ability to rotate the entire manipulator around its vertical axis the six motors five motors for each joint angle and one motor to rotate the entire manipulator around its vertical axis of the manipulator should be controlled Using the method of our papers 12 13 the configuration of the manipulator will have three angles to be controlled instead of n angles Figure 1 b shows the configuration of the manipulator when there are just three angles that need to be controlled Because the end effector can follow any desired path by controlling three angles 1 2 and 3 only therefore instead of using a motor for each joint angle three motors can be used for controlling the manipulator This means that for any number of degrees of freedom three dimensional planar redundant manipulators the weight of the links will be significantly decreased using the proposed design To make the manipulator capable of moving in a three dimensional work space one motor will control the value of 1 this means controlling the rotation of the entire manipulator around the vertical axis This motor is situated in such a way as to rotate the base of the manipulator around the z axis The second motor controls the value of 2 which means the rotation of the entire manipulator with its configuration The motor is situated at the base The third motor controls the value of 3 and this motor is situated on the first link This motor will rotate the second link of manipulator about the second axis and because all the next links should rotate about their axes by the same angle 3 therefore there is no need to use motor for each Sensors 2012 12 6872 joint angle but the rotation of the second motor will be transferred to the next joints using gears boxes Figure 2 shows the mechanism of the proposed manipulator Figure 1 a A three dimensional planar redundant manipulator configuration b A three dimensional planar redundant manipulator configuration using the method of 12 13 a b Figure 2 The manipulator used in experiments 14 The draft of the manipulator using the SolidWorks software left The mechanical design of the manipulator right Elaborating further the second motor is connected to the first link using a worm gear to control the angle 2 Figure 3 shows the position of the second motor l n l 3 l 1 0 0 l 2 y axis y tp z tp x tp x tp y tp z tp s x axis z axis 3 4 n 1 2 1 l n l 3 l 1 0 0 l 2 y axis y tp z tp x tp x tp y tp z tp s x axis z axis 3 3 3 2 1 Second motor Third motor First motor Sensors 2012 12 6873 Figure 3 The design of the second joint angle first link with second motor of the manipulator 14 The draft of the second joint angle using the SolidWorks software left The mechanical design of the second joint angle right 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 i e the angle 3 Figure 4 shows the position of the third motor Figure 4 The design of the third joint angle second link with third motor of the manipulator 14 The draft of the third joint angle using the SolidWorks software top left The draft of the whole manipulator using the SolidWorks software top right The mechanical design of the third joint angle bottom The mechanism of the third link is shown in Figure 5 The same mechanism of the second link is used the only one difference is that instead of using s worm as a driver and s wheel gear as a driven two bevel gears are used The same mechanism of the third link can be used with the next links The last link has the mechanism shown in the Figure 6 For further details of the mechanical design of the manipulator our reference 14 can be checked worm driver gear 1 st link Second motor worm gear wheel planetary gear bevel gear 1 bevel gear 2 arm 1 st link 2 nd link Third motor Sensors 2012 12 6874 Figure 5 The design of the fourth joint angle third link of the manipulator 14 The draft of the fourth joint angle using the SolidWorks software top left The draft of the whole manipulator using the SolidWorks software top right The mechanical design of the fourth joint angle bottom Figure 6 The last joint of the manipulator 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 1 2 2 1 N N w w 1 the last link Sensors 2012 12 6875 where w is the angular velocity of gear and N is the number of teeth of gear In our manipulator it is noted that the first gear is fixed while the second gear and the arm are rotating It is desired that both the arm and the second gear have the same angular velocity Because the arm is not stationary then we cannot use the previous equation i e the mechanism is not an ordinary gear train but a planetary gear train To convert this planetary gear train to an ordinary gear train it is assumed that the arm is stationary while a first gear has an angular velocity and not fixed This means that a www 11 2 0 aaa www 3 And because the second gear will continue rotating with the same angular velocity then 22 ww 4 Now the Equation 1 can be rewritten as follows 2 1 1 2 2 1 w ww N N w w a 5 For our manipulator it is desired to move both the arm and the second gear by the same angular velocity w which means 1 2 0 N N w w 21 NN 6 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 z axis This motor controls 1 Figure 7 shows the mechanism of the first motor Figure 7 The mechanism of the first motor To calculate the transformation matrix of the manipulator the draft of the manipulator shown in Figure 8 is used The corresponding link parameters of the manipulator are shown in Table 1 Where l 1 l 2 l 5 are the length of the links while d 1 is the offset between the origin and the end effector Sensors 2012 12 6876 Figure 8 The manipulator used in experiments Table 1 Link parameters of the manipulator i a d 1 90 0 0 1 2 0 l 1 d 1 2 3 0 l 2 0 3 4 0 l 3 0 4 5 0 l 4 0 5 6 0 l 5 0 6 From the links parameters shown in Table 1 and using Equation 7 which defines the transformation matrix T for the links 1 we compute the individual transformations for each link 1000 cossin 0 sinsincoscoscossin cossinsincossincos 1 d a a T iii iiiiiii iiiiiii i i 7 where c i cos i and s i sin i 1000 0010 00 00 11 11 1 0 cs sc T 1000 100 0 0 1 2122 2122 2 1 d slcs clsc T 1000 0100 0 0 3233 3233 3 2 slcs clsc T 1000 0100 0 0 4344 4344 4 3 slcs clsc T 1000 0100 0 0 5455 5455 5 4 slcs clsc T 1000 0100 0 0 6566 6566 6 5 slcs clsc T 8 Finally we obtain the product of all six link transforms 6 5 5 4 4 3 3 2 2 1 1 0 6 0 TTTTTTT 9 x axis y axis z axis d 1 l 1 l 2 l 3 l 4 l 5 origin target point Sensors 2012 12 6877 3 Dynamics of the Manipulator 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 ml 1 760 gm ml 2 720 gm ml 3 680 gm ml 4 640 gm and finally ml 5 600 gm These masses are calculated for the manipulator with l 1 19 cm l 2 18 cm l 3 17 cm l 4 16 cm l 5 15 cm and d 2 21 cm 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 m 1 2 260 gm m 2 720 gm m 3 680 gm m 4 640 gm and m 5 600 gm Figure 9 a shows the mass of each link with its motor for the manipulator of the proposed design Figure 9 The position of mass for a the proposed manipulator b the conventional manipulator 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 i e 760 1 500 gm The mass of the second link will equal to the mass of link itself plus the mass of the motor which controls the third link position i e 720 1 500 gm The mass of the third link will equal to the mass of third link plus the mass of the motor which controls the fourth link position i e 680 1 500 gm The mass of the fourth link will equal to the mass of fourth link plus the mass of the motor which controls the fifth link position i e 640 1 500 gm while the mass of the last link will equal to the mass of the link itself because there are no more motors i e 600 gm Figure 9 b shows the mass of each link using the manipulator with five motors while Table 2 shows the values of mass of the links using both the manipulator with two motors and the manipulator with five links x axis y axis z axis ml 2 ml 3 ml 4 ml 5 origin target point m motor2 ml 1 m motor3 x axis y axis z axis origin target point m motor2 ml 1 m motor3 ml 5 ml 3 m motor5 ml 4 m motor6 ml 2 m motor4 Sensors 2012 12 6878 Table 2 The mass of links for both the proposed and conventional manipulators m n gm Proposed manipulator Conventional manipulator m 1 gm 1 500 1 500 m 2 gm 2 260 2 260 m 3 gm 720 2 220 m 4 gm 680 2 180 m 5 gm 640 2 140 m 6 gm 600 600 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 6 1j iiijij QGVqM 10 for i 1 2 6 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 1 20 21 Dynamics equations of the manipulator are discussed in details in the Appendix 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 22 the parallel manipulator is actuated by three servo motors located at the base which contributes to reducing the inertia of manipulators Reference 23 shows another way to decrease the effect of the motors weight on the inertia of manipulators This reference shows a simple configuration design which comprises of only three joints two at the shoulder and one at the hand In this design the moment of inertia of the arm is constant and independent from the joint angles In contrast for our manipulator we see from Equations A21 A25 that the moment of inertia value is dependent on the joint angles 4 Simulation Results 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