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International Journal of Engineering and Technology Volume 2 No. 10, October, 2012 An Inverse Kinematic Analysis of a Robotic Sealer Akinola A. Adeniyi 1, Abubakar Mohammed 2, Aladeniyi Kehinde 3 1Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria 2Department of Mechanical Engineering, Federal University of Technology, Minna, Nigeria 3Department of Science Laboratory Technology, Rufus Giwa Polytechnic, Owo, Nigeria ABSTRACT A planar robotic sealing or brand stamping machine is presented for an automated factory line. The appropriate time to seal or to stamp an object is basically determined by a motor controller which relies critically on whether or not the object is in the best position. The extent of protraction and retraction of the piston head is largely dictated by an infrared sensor. Given the extent to protract or retract the piston head, the angular displacements of the link required are determined using the Inverse Kinematic (IK) techniques. The inertia and gravity effects of the links have been ignored to reduce the complexity of the equations and to demonstrate the technique. Keywords: Forward Kinematics, Inverse Kinematics, Robotics, Sealer 1. INTRODUCTION An automated factory uses a number of mechanical links electronically controlled to achieve tasks. The benefits of factory automation are many and of strategic importance to management [1]. Standard mechanical links are usually powered with electrical motors, pneumatic systems or solenoids. In a manually operated machine, the human performs visual checks and other standard checks that are to be replicated by automation. The interest of this work is centered on a hypothetical sealing machine which is used for stamping some signatures and logos as done in a branding factory line. Inverse kinematic analysis is applied to enable us determine angular displacements of the link. Kinematics involves the study of motion without consideration for the actuating forces. Inverse Kinematics (IK) is a method for determining the joint angles and desired position of the end-effectors given a desired goal to reach by the end effectors [1]. A feasibility of using a PID controller was studied by Nagchaudhuri [2] for a slider crank mechanism but without an offset. Tolani et al [3] reviewed and grouped the techniques of solving inverse kinematics problems into seven. The techniques are the Newton-Raphson’s method and its other variants. There are the Jacobian and the variants with pseudo-inverse (otherwise known as the Moore-Penrose inverse) for square or non-square Jacobian. Other methods are the control-theory based and the optimisation techniques. A number of authors [1, 4-7] have proposed algorithms for solving IK problems which include but not limited to Neural Network algorithm, Cyclic Coordinate Descent closure and Inexact strategy, but like every other techniques for a given problem the choice of method depends on the specifics of the problem. Buss [8] discussed the Jacobian transpose, the Moore-Penrose and the Damped Least Squares techniques. In terms of computational cost, the Jacobian transpose method is the cheap but can perform poorly based on the robot configurations. In this work the Jacobian transpose technique ill-performed but the Jacobian Inverse technique is suitable and more so it is a simple 2D planar representation of the problem with only 4 degrees of freedom. 2. OPERATIONS OF THE ROBOTIC LINK Fig. 1 shows the schematic diagram of the robotic sealing system. The capping or stamping is achieved with the piston or ram head, P. C is the conveyor line. The caps or the branding heads are placed in position and sensed by an infrared sensor, S. The instruction to seal or brand is dependent on feedback from the sensor. If the item to be branded, capped or stamped is out of place at the instance when the ram head was going to touch, the sensor feedback will be to retract the head. It can also be to not go too far. There can be a range of feedback to the motor controller, M. This kind of control system is similar to what a human operator would do if it were manually operated. The use of sensors and fast responding motor controller will make this hypothetical machine a very useful tool in a factory performing this kind of mundane task. This factory sub-line is a simple slider-crank mechanism with actuator arm A. In clearer terms, the instructions would be to press the piston ram to seal if the cap and the container are in line; to reverse the piston in case of a jam; to not press the piston ram if either the container or the cap is absent; to press further if the seal length is shorter than expected as may be caused by wear and tear. This clearly shows that the piston determines the angle of the link or the direction or action of the motor. This is an inverse kinematics problem. The sensor feedback part is much of a control engineering problem, not considered in this paper. Fig. 1: The robotic sealing rig schematic 3. ANALYSIS Fig. 2 is a representation of the slider-crank mechanism. There is an offset, f, of the piston axis from the motor axis, O1. O2 is the axis of the piston with moving coordinates (x,y). The motor rotates clockwise or counter clockwise about O1. If the crank makes displacement Δs on the piston plane, it is equivalent to a motion of Δex and Δey. This motion is caused by the crank making an angular motion clockwise or counter-clockwise, . The angle between the connecting rod and crank makes an angular displacement of, . This also means the angular shift of is made between the connecting rod and the piston or ram plane. Fig. 2: The offset slider crank (Cartesian coordinate world) In a computer game application for these, the angles would be explicitly required so that the links do not “physically disjoint”; for a physically connected link, the motor controller only would need the instruction to move only the crank. 3.1 The World Cartesian coordinate system is adopted. Clockwise is positive and motion to right and upwards are positive. The Top Dead Centre (TDC) is attained when the crank, radius r, and the connecting rod, length l, are in line. This is attained when . fm is the maximum variable offset based on the geometry. The Bottom Dead Centre (BDC) is reached when . The TDC and BDC with the variable offset are shown in Fig. 3. Fig. 3: The Top and Bottom Dead centre The piston has been constrained to move only in planar direction, on the vector of . In this work, the direction vector is , making the plane at 45° to the horizontal. 3.2 The Forward Kinematics The displacement caused by the motor moving clockwise from the position in Fig. 2 is represented in equation (1). Where subscripts (i,f) are respectively mean initial and final values. The position at f is reached in reality smoothly for a rotating crank, but the smoothness can be reached in fine incremental steps, in the numerical approach. At the end of the stepped increments, the final displacement to the goal is seen as a function of angular parameters given as: The linear dependence of the angles, in this problem, can help to reduce the number of degrees of freedom to compute in equation (1). It can be shown that, there by making .Using trigonometry, the instantaneous initial, arbitrary, position of the piston in Fig. 2 is given by Equation (2)(3). The Jacobian matrix for is given in equation (4) and simplified to equation (5). Computing the new piston position involves solving equation (1). The new coordinate of the piston by the first term of expansion of the Taylor series can be shown to be given in equation (6). θis the vector of the robot angular displacements for the related links. Mathematically, . Here, we have . Therefore the current position of the piston or the pressing head is approximately given in equation (6). It should be noted thatβcan be measured from the horizontal to further reduce the equation sets, this is referred to as β0 elsewhere in this paper. 3.3 Inverse Kinematics The problem is not that of solving for Xf given Xi andθbut it is that of solving for θgiven Xi, and the desired Xf. This is iteratively implemented such that the target displacement of the piston is given as .This is a vector of the piston displacement and can be represented as.Since this is a planar problem with no displacements in the other directions, it reduces to a.To smoothen the possible jerk or jumpy effect, this can be stepped using a factor ofμwhich can be selected intuitively based on the ratio of r to L but and is the inverse of Jacobian matrix. The algorithm checks if the target has been reached or not. Iteration is stopped when the solution is within a pre-determined level of error or a maximum number of iterations. The choice of these limiting values should depend on the response time acceptable. This can be critical for a real time application. 4. RESULT AND DISCUSSIONS Consider a current orientation of the robotic arm at any arbitrary position with the piston head at a position P. Suppose the sensor system requires the piston to move to a target new position P2. The simulation is done for several arbitrary starting positions of the crank and results are similar for reachable targets. Supposing the crank angle is at a current orientation with crank angle of -5°, and there is an instruction from the sensor to retract the piston ram head by 0.1times the crank arm length. The simulation instructs the crank proceeds to counter clockwise by 15.58°, this corresponds to an increase ofβ0 to 19.26°and correspondingly,λreduces to 86.32°. Fig. 4 shows the simulation progress of the piston head from a current position P1 to the new target P2 and the number of iterations done. Fig. 4: Crank Position and Iteration with the Jacobian Inverse Matrix The technique used is the Jacobian inverse technique. The Jacobian transpose technique is not predictable for the same problem and in this case, the solution settles to a local minimum for only one of the angles but the convergence rate is faster, see Fig. 5. Fig. 5: Crank Positions using the Inverse and Transpose of the Jacobian Matrix If there is a request to a physically unreachable target, such as to a more than the TDC or BDC locations, P3, the simulation runs and stops after the maximum number of iterations or if the Jacobian Matrix becomes un-invertible, Fig. 6. Fig. 6: Unreachable Target situation 5. CONCLUSION This paper is focused on the application of the Inverse Kinematics technique to the analysis of a robotic link, such as obtained in a sealer of an automated factory, without consideration for the effects of inertia effects. The Jacobian inverse technique, as mentioned in literatures, is more reliable in this application. The Jacobian transpose approach is not reliable. This paper has demonstrated the application of the inverse kinematics to a simple robotic sealer; the piston is instructed to retract by 0.1 units as a test case. The new crank angle was found more accurately with the Jacobian Inverse technique better that the Jacobian Transpose technique. The problem can be extended to include the dynamics for possible selection of the optimal driving torque or electric motor selection for the driving parts. REFERENCES [1] S. Tejomurtula and S. Kak, "Inverse Kinematics in robotics using neural networks," Information Sciences, vol. 116, pp. 147-164, 1999. [2] A. Nagchaudhuri, "Mechantronic Redesign of Slider Crank Mechanism," in ASME International Mechnical Engineering Congress & Exposition: IMECE2002, New Orleans, Louisiana, 2002. [3] D. Tolani, A. Goswami, and N. I. Badler, "Real-Time Inverse Kinematics Techniques for Anthromorphic Limbs," Graphical Models, vol. 62, pp. 353-388, 2000. [4] S. K. Saha and W. O. Schiehlen, "Recursive Kinematics and Dynamics for Parallel Structured Closed-Loop multibody Systems," Mechanics of Structures and Machines, vol. 29, pp. 143-175, 2007. [5] X. Wang, "A behavior-based inverse kinematics algorithm to predict arm prehension postures for computer-aided ergonomic evaluation," Journal of Biomechanics, vol. 32, pp. 453-460, 1999. [6] A. C. Nearchou, "Solving the inverse kinematics problem of redundant robots operating in complex environments via a modified genetic algorithm," Mechanism and Machine Theory, vol. 33, pp. 273-292, 1998. [7] M. J. D. Powell, "Some Global Convergence Properties of a variable metric Algorithm for Minimization without Exact line searches," in Symposium in Applied Mathematics of the American Mathematical Society and the Society for Industrial and Applied Mathematics, New York City, 1976. [8] S. R. Buss, "Introduction to Inverse Kinematics with Jocobian Transpose, Pseudoinverse and Damped Least Square methods," University of California, San Diego2009. 課題內(nèi)容 本課題的內(nèi)容是關(guān)于塑料板材折彎機的設(shè)計，主要包括塑料折彎機的結(jié)構(gòu)設(shè)計以及對現(xiàn)有產(chǎn)品的結(jié)構(gòu)優(yōu)化和創(chuàng)新。能夠?qū)崿F(xiàn)較高折彎精度和加工效率，例如，1）設(shè)計多功能的折彎模具或組合刀架，減少換刀次數(shù)；2）優(yōu)化結(jié)構(gòu)減少加熱塑料板材的時間，提高工作效率；3）設(shè)計簡易的上下料裝置和板材對齊裝置，提高加工質(zhì)量和工作效率。主要研究對象是手動或半自動折彎機的結(jié)構(gòu)設(shè)計，通過Solidworks等三維建模軟件對設(shè)計的折彎機進(jìn)行三維造型并仿真，利用有限元分析等計算合理的折彎力等。分析工作塑料板材的熱彎特性等，設(shè)計合理的加熱裝置。然后再將三維造型圖轉(zhuǎn)化為工程圖，并且計算確定具體結(jié)構(gòu)的參數(shù)，同時完成零件圖的繪制和相關(guān)標(biāo)注等。最后完成對本課題的論文撰寫。 課題任務(wù)要求 1.查閱塑料折彎機相關(guān)文獻(xiàn)并進(jìn)行收集和整理，完成畢業(yè)設(shè)計前期工作：任務(wù)書、文獻(xiàn)綜述、開題報告、外文翻譯； 2.查閱大量關(guān)于折彎機結(jié)構(gòu)設(shè)計的資料、塑料熱成型相關(guān)資料。整理后確定塑料折彎機的具體設(shè)計方案； 3.進(jìn)行必要的設(shè)計計算和結(jié)構(gòu)設(shè)計，如折彎系數(shù)、折彎力等計算，凸模凹模的結(jié)構(gòu)和尺寸的設(shè)計。保證具備良好的機械性能； 4.運用CAD等軟件，對所設(shè)計的塑料折彎機進(jìn)行三維建模，繪制裝配圖，零件圖以及重要機構(gòu)的運動仿真，并進(jìn)行有限元分析； 5利用三維建模完成工程圖的轉(zhuǎn)換，工程圖需要體現(xiàn)出具體結(jié)構(gòu)的尺寸參數(shù)等； 6.最終設(shè)計出的塑料折彎機要具備比現(xiàn)有同類產(chǎn)品更方便高效的性能，同時保證加工精度不低于現(xiàn)有的同類產(chǎn)品。 主要參考文獻(xiàn)（由指導(dǎo)教師選定） [1] 孫靖民 梁迎春.機械優(yōu)化設(shè)計 第四版[M].北京：機械工業(yè)出版社，2006 [2] 孫桓 陳作模 葛文杰.機械原理 第七版[M].高等教育出版社，2006 [3] 齊民.機械工程材料 第七版[M].大連：大連理工大學(xué)出版社，2007 [4] 李華志.數(shù)控加工工藝與裝備[M].北京:清華大學(xué)出版社,2005. [5] 林清安.Pro/ENGINEER 2001鈑金設(shè)計[M].北京:清華大學(xué)出版社,2001 [6] 彭詩林等.鈑金件結(jié)構(gòu)設(shè)計工藝手冊[M] [7] 魏龍祥.W67K—400/4000型數(shù)控板料折彎機簡介[J].上海機床,1997,02期 [8] 哈爾濱工業(yè)大學(xué)理論力學(xué)教研室.理論力學(xué)I 第七版[M].高等教育出版社，2004 [9] 伍時雄.塑料板折彎機成型裝置[P].中國:00267582.X,2001 [10] 劉世全.塑料板折彎機[P].中國:CN 201446703 U,2009 [11] 劉鴻文.材料力學(xué)I 第四版[M].高等教育出版社，2004 同組設(shè)計者 無 注：1、任務(wù)書由指導(dǎo)教師填寫；2、任務(wù)書在第七學(xué)期第15周前下達(dá)給學(xué)生。 學(xué)生完成畢業(yè)設(shè)計（論文）工作進(jìn)度計劃表 序號 畢業(yè)設(shè)計（論文）工作任務(wù) 工 作 進(jìn) 度 日 程 安 排 周次 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 查找整理課題相關(guān)資料文獻(xiàn)等 — — 2 撰寫開題報告、文獻(xiàn)綜述、任務(wù)書 — — — — 3 閱讀英文文獻(xiàn)并翻譯一篇英文文獻(xiàn) — — 4 前期工作的匯總整理 — — 5 確定具體方案以及整理思路 — — — 6 三維建模、完成造型設(shè)計 — — 7 進(jìn)行裝配及運動仿真 — — — 8 三維轉(zhuǎn)換為工程圖 — — — 9 完成課題論文的撰寫 — — — — — 注：1、此表由指導(dǎo)教師填寫； 2、此表每個學(xué)生一份，作為畢業(yè)設(shè)計（論文）檢查工作進(jìn)度之依據(jù)； 3、進(jìn)度安排用“—”在相應(yīng)位置畫出。 畢業(yè)設(shè)計（論文）階段工作情況檢查表 時間 第 一 階 段 （撰寫開題報告、外文翻譯、文獻(xiàn)綜述） 第 二 階 段 （論文初稿撰寫或方案設(shè)計） 第 三 階 段 （論文終稿或圖紙繪制） 內(nèi)容 組織紀(jì)律 完 成 任 務(wù) 情 況 組織紀(jì)律 完 成 任 務(wù) 情 況 組織紀(jì)律 完 成 任 務(wù) 情 況 檢 查 情 況 教師簽字 簽字 日期 簽字 日期 簽字 日期 注：1、此表由指導(dǎo)教師認(rèn)真填寫； 2、“組織紀(jì)律”一欄根據(jù)學(xué)生具體執(zhí)行情況如實填寫； 3、“完成任務(wù)情況”一欄按學(xué)生是否按進(jìn)度保質(zhì)保量完成任務(wù)的情況填寫； 4、對違紀(jì)和不能按時完成任務(wù)者，指導(dǎo)教師可根據(jù)情節(jié)輕重對該生提出警告或不能參加答辯的建議。