立式精鍛機(jī)自動(dòng)上料機(jī)械手手部結(jié)構(gòu)的設(shè)計(jì)含12張CAD圖.zip,立式,精鍛機(jī),自動(dòng),機(jī)械手,結(jié)構(gòu),設(shè)計(jì),12,CAD 關(guān)節(jié)轉(zhuǎn)矩降低三維冗余的平面機(jī)械手 1.研究中心在吉隆坡50603馬來西亞大學(xué)應(yīng)用電子科，馬來西亞;電子郵件:mahmoud@um.edu.my。 2.電氣和計(jì)算機(jī)工程學(xué)院,德黑蘭大學(xué),郵政信箱14399 - 57131。 3.電氣與電子工程系：馬來西亞諾丁漢大學(xué),道路Broga,Semenyih43500；馬來西亞;電子郵件:haider.abbas@nottingham.edu.my。 摘要 研究機(jī)器人機(jī)械手關(guān)節(jié)力矩的還原已經(jīng)受到近年來相當(dāng)多的關(guān)注。其可以減少計(jì)算復(fù)雜度的扭矩，優(yōu)化計(jì)算能力的大小使關(guān)節(jié)力矩準(zhǔn)確地將結(jié)果體現(xiàn)在在一個(gè)安全的操作,沒有超載的聯(lián)合執(zhí)行機(jī)構(gòu)。本文提出：機(jī)械設(shè)計(jì)的三維平面冗余機(jī)械手，利用數(shù)目的減少確定需要控制關(guān)節(jié)角，領(lǐng)先在機(jī)械手的重量的減少。 許多的努力都集中在減少機(jī)械手的重量,如使用輕量級(jí)關(guān)節(jié)設(shè)計(jì)或設(shè)置執(zhí)行器在機(jī)械手的基礎(chǔ)和使用肌腱的動(dòng)力傳遞到這些關(guān)節(jié)。通過本文的設(shè)計(jì)，只有三個(gè)電動(dòng)機(jī)需要控制任何n度自由在一個(gè)三維平面冗余機(jī)械手臂代替n電機(jī)自由。因此，這個(gè)設(shè)計(jì)是為了減少機(jī)器人的重量以及一些非常有效的需要控制的機(jī)械手電機(jī)。在本文中，所有的關(guān)節(jié)力矩提出了計(jì)算該機(jī)械手（三個(gè)馬達(dá)）和傳統(tǒng)的三維平面機(jī)械手（有一個(gè)電機(jī)，每個(gè)自由度）來顯示，機(jī)械手的有效性提出減少的重量和機(jī)械手驅(qū)動(dòng)關(guān)節(jié)磨損最小化。 關(guān)鍵詞：冗余機(jī)械手;動(dòng)力學(xué);機(jī)器人;旋轉(zhuǎn)編碼器;減少關(guān)節(jié)磨損 1 介紹 從理論上講，對(duì)于一個(gè)機(jī)器人機(jī)械手的結(jié)構(gòu)可以安裝在每個(gè)環(huán)節(jié)通過減速器驅(qū)動(dòng)下一個(gè)鏈接，但執(zhí)行機(jī)構(gòu)和減速器安裝在末端成為負(fù)載執(zhí)行機(jī)構(gòu)安裝在近端結(jié)束的機(jī)械手,導(dǎo)致一個(gè)笨重的超重系統(tǒng)[1]。為了減輕重量和機(jī)器人慣性，到目前為止，機(jī)制已經(jīng)提出去除重量的限制。一些報(bào)道[2，3]包括： （一）輕量級(jí)聯(lián)合設(shè)計(jì)基于一個(gè)特殊的旋轉(zhuǎn)接頭(4 - 6) （二）在一個(gè)滑塊底部承受盡可能多的所需驅(qū)動(dòng)力[7] （三）并行機(jī)制是另一種方法來減少質(zhì)量和慣性的機(jī)械手[8] 一個(gè)典型的機(jī)并聯(lián)械手是由一個(gè)移動(dòng)平臺(tái)與固定座連接，一般說,自由度的數(shù)目與并聯(lián)機(jī)械手的四肢的數(shù)量相等。該驅(qū)動(dòng)器通常是安裝在基地附近,這有助于減少機(jī)器人的慣性。 （四）濃度的致動(dòng)器在每個(gè)接頭的基礎(chǔ)和動(dòng)力傳輸通過肌鍵的一個(gè)特殊的傳動(dòng)機(jī)構(gòu)[2、3、9]。這種機(jī)制允許執(zhí)行機(jī)構(gòu)是位于遠(yuǎn)程操作器上的基地,使機(jī)械手做的更輕和緊湊。 對(duì)于一系列機(jī)械手,直接運(yùn)動(dòng)學(xué)是相當(dāng)簡單的,而逆運(yùn)動(dòng)學(xué)變得非常困難。參考[10]提出了一種融合智能傳感器網(wǎng)絡(luò)估計(jì)了工業(yè)機(jī)器人運(yùn)動(dòng)學(xué),同時(shí)參考[11]措施范圍的數(shù)據(jù)關(guān)于機(jī)器人基礎(chǔ)構(gòu)架使用機(jī)器人正運(yùn)動(dòng)學(xué)和光學(xué)三角測量原理。逆運(yùn)動(dòng)學(xué)問題是更有趣的是和它的解決方案是更有用的,但是遇到的問題之一逆運(yùn)動(dòng)學(xué)是當(dāng)機(jī)械手是冗余的,預(yù)計(jì)的逆運(yùn)動(dòng)學(xué)有無窮多組解。這意味著,對(duì)于一個(gè)給定位置的機(jī)械手的端部執(zhí)行器,就有可能誘發(fā)一個(gè)自動(dòng)的結(jié)構(gòu)而不改變位置的端部執(zhí)行器。在本文中,我們依靠我們現(xiàn)有的作品[12,13],提出一個(gè)新的方法來解決—三元的平面冗余機(jī)械手臂的多解問題。由于本文闡述了機(jī)械手的動(dòng)態(tài)而不是他的運(yùn)動(dòng)學(xué)，逆運(yùn)動(dòng)學(xué)方法將無法解釋這里。關(guān)于逆運(yùn)動(dòng)學(xué)冗余機(jī)械手的更多細(xì)節(jié),我們的作品[14]可以檢查。 這是前面提到的,該機(jī)械手可以用來減少的重量，機(jī)械手收益率下降用于控制機(jī)械手的電機(jī)。在降低動(dòng)電動(dòng)機(jī)的反力矩上顯示了該機(jī)械手的有效性，對(duì)動(dòng)態(tài)的機(jī)械手進(jìn)行了數(shù)學(xué)計(jì)算。逆動(dòng)態(tài)模型提供了在術(shù)語的關(guān)節(jié)磨損的關(guān)節(jié)位置、速度和加速度。為機(jī)器人設(shè)計(jì),逆動(dòng)態(tài)模型是用來計(jì)算執(zhí)行機(jī)構(gòu)的扭矩,以獲得所需的運(yùn)動(dòng)[17]。幾種方法已經(jīng)提出的動(dòng)力學(xué)模型。最常用于機(jī)器人的拉格朗日公式和牛頓歐拉公式。由于拉格朗日公式在概念上很簡單,系統(tǒng)[18],它被用在這里。拉格朗日公式提供了一個(gè)描述聯(lián)合執(zhí)行機(jī)構(gòu)之間的關(guān)系力量和運(yùn)動(dòng)的機(jī)制,在系統(tǒng)里從根本上操作動(dòng)能和勢(shì)能。 本文的工作是基于先前的工作[14],呈現(xiàn)了一個(gè)機(jī)械設(shè)計(jì)一個(gè)三維平面冗余機(jī)械手,保證減少重量的機(jī)械手通過減少需要控制汽車的數(shù)量的機(jī)械手。因?yàn)槟孢\(yùn)動(dòng)學(xué)模型提供了一個(gè)無限數(shù)量的解決方案為冗余機(jī)械手,因此,二次性能標(biāo)準(zhǔn)可以優(yōu)化[17],如避免奇異構(gòu)型和最小化驅(qū)動(dòng)關(guān)節(jié)力矩。參考[14]研究了本文的機(jī)械手運(yùn)動(dòng)學(xué)和顯示在細(xì)節(jié)能夠避免奇異的配置能力。可操縱性指數(shù)數(shù)值和可操縱性橢圓體的機(jī)械手是由索引值的可操縱性和彪馬的可操縱性橢圓體手臂顯示使用的有效性提出了機(jī)械手,以避免奇異機(jī)械手。在本文中,對(duì)機(jī)械手的動(dòng)力學(xué),進(jìn)行了詳細(xì)解釋。這個(gè)工作的貢獻(xiàn)是解釋該機(jī)械手的關(guān)節(jié)力矩極小化的能力。研究了該鏈接和電機(jī)的質(zhì)量分布(三個(gè)馬達(dá))和傳統(tǒng)的機(jī)械手(汽車)。驅(qū)動(dòng)關(guān)節(jié)磨損研究提出的機(jī)械手各關(guān)節(jié)和結(jié)果，與傳統(tǒng)的機(jī)械手結(jié)果比較表明該機(jī)械手的有效性的最小化為驅(qū)動(dòng)關(guān)節(jié)磨損。 2 機(jī)械手的機(jī)械設(shè)計(jì) 控制運(yùn)動(dòng)圖1顯示的機(jī)械手的末端執(zhí)行器的運(yùn)動(dòng)（一）,所有的汽車機(jī)械手應(yīng)受控制。例如,控制五個(gè)環(huán)節(jié)平面冗余機(jī)械手能夠旋轉(zhuǎn)整個(gè)機(jī)械手在其垂直軸的能力,六個(gè)馬達(dá)(五發(fā)動(dòng)機(jī)各關(guān)節(jié)角和一個(gè)電動(dòng)機(jī)旋轉(zhuǎn)整個(gè)機(jī)械手在其垂直軸)應(yīng)該控制的機(jī)械手。使用該方法的論文[12,13],配置的機(jī)械手將有三個(gè)角度控制代替n角度。圖1(b)顯示了配置機(jī)械手在只有三個(gè)角度時(shí),需要控制。 因?yàn)槟┒丝梢宰裱魏蜗胍穆窂酵ㄟ^控制三個(gè)角度(θ1,θ2和θ3),因此,不使用電動(dòng)機(jī)為每個(gè)關(guān)節(jié)角,三個(gè)汽車可以用于的控制機(jī)械手。這意味著對(duì)于任何數(shù)量的自由度三維平面冗余的權(quán)重,鏈接的重量將會(huì)明顯降低使用提出的設(shè)計(jì)。讓能夠移動(dòng)的機(jī)械手在三維工作空間,一個(gè)馬達(dá)控制的價(jià)值的θ意味著控制整個(gè)機(jī)械手的旋轉(zhuǎn)圍繞垂直軸。這個(gè)電機(jī)坐落在這樣旋轉(zhuǎn)底部的機(jī)械手在z軸 。第二個(gè)電動(dòng)機(jī)控制θ2的值,這意味著整個(gè)機(jī)械手的旋轉(zhuǎn)與它的配置。這個(gè)電動(dòng)機(jī)位于該基地。第三個(gè)電動(dòng)機(jī)控制θ3的價(jià)值,這種電動(dòng)機(jī)位于第一個(gè)鏈接。 這種電動(dòng)機(jī)將旋轉(zhuǎn)臂的第二個(gè)鏈接關(guān)于第二軸,因?yàn)樗械南乱粋€(gè)鏈接應(yīng)該對(duì)他們的軸旋轉(zhuǎn)相同的角θ3。因此,沒有必要使用電動(dòng)機(jī)為每個(gè)關(guān)節(jié)角,但第二電機(jī)的旋轉(zhuǎn)將被轉(zhuǎn)移到下一個(gè)關(guān)節(jié)使用齒輪箱。圖2顯示了該機(jī)械手的機(jī)構(gòu)。三維平面冗余機(jī)械手配置;(b)三維平面冗余機(jī)械手配置使用方法[12,13]。 用于實(shí)驗(yàn)的機(jī)械手[14]。草案操縱者利用SolidWorks軟件(左)。機(jī)械設(shè)計(jì)的機(jī)械手(右)。進(jìn)一步闡述,第二馬達(dá)連接到第一個(gè)鏈接使用一個(gè)蝸輪控制角θ2。顯示了第二電動(dòng)機(jī)的位置。第二個(gè)關(guān)節(jié)角的設(shè)計(jì)(第一個(gè)鏈接與第二電動(dòng)機(jī))[14]。利用SolidWorks軟件的二關(guān)節(jié)節(jié)角草案(左)。第二個(gè)關(guān)節(jié)角的機(jī)械設(shè)計(jì)(右)。第三個(gè)馬達(dá)連接到第二個(gè)鏈接使用一個(gè)蝸輪因?yàn)橥瑯拥脑蚴鞘褂玫谝粋€(gè)鏈接。 控制第三汽車意味著控制角之間的第一個(gè)和第二個(gè)鏈接即鏈接。,θ3角度。第三關(guān)節(jié)角機(jī)械手的設(shè)計(jì)(第二個(gè)鏈接與第三電動(dòng)機(jī))[14]。草案第三關(guān)節(jié)角利用SolidWorks軟件(左上角)。利用SolidWorks軟件對(duì)整個(gè)機(jī)械手的草案(右上)。第三個(gè)關(guān)節(jié)角度的機(jī)械設(shè)計(jì)(底部)。 相同的機(jī)制的第二個(gè)鏈接使用;唯一的區(qū)別是,而不是使用年代蠕蟲作為驅(qū)動(dòng)的齒輪為驅(qū)動(dòng),使用兩個(gè)錐齒輪。第三連桿的相同機(jī)制可用于下一個(gè)鏈接。最后的鏈接有機(jī)制如圖6所示。為進(jìn)一步的細(xì)節(jié)的機(jī)械設(shè)計(jì)的機(jī)械手,我們參考[14]可以檢查。 確保所有的鏈接而運(yùn)動(dòng)的關(guān)節(jié)角,錐齒輪之間的比例每個(gè)行星齒輪應(yīng)該等于一。這意味著錐齒輪的每個(gè)行星齒輪應(yīng)該有相同的直徑和數(shù)量的牙齒。 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