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重 庆 理 工 大 学文 献 翻 译二级学院 重庆汽车学院 班 级 0940205 学生姓名 毛奕 学 号 10904020317译 文 要 求1、译文内容必须与课题（或专业）内容相关，并需注明详细出处。2、外文翻译译文不少于2000字；外文参考资料阅读量至少3篇（相当于10万外文字符以上）。3、译文原文（或复印件）应附在译文后备查。 译 文 评 阅导师评语（应根据学校“译文要求”，对学生外文翻译的准确性、翻译数量以及译文的文字表述情况等作具体的评价） 指导教师： 年 月 日国际工程与技术杂志2卷10号，十月，2012自动封口机的逆运动学分析Akinola A. Adeniyi 1, Abubakar Mohammed 2, Aladeniyi Kehinde 3机械工程1，大学管理，管理，尼日利亚机械工程系，联邦技术大学，明娜，尼日利亚科学实验室技术系，鲁弗斯性理工，尼日利亚摘要自动封口机主要用于自动化生产线。正确封装一样物品的时间基本取决于电机控制器是否能精确地判断出这样物品在不在最佳的位置上。活塞头的伸缩度很大程度上取决于红外传感器，由活塞头的伸缩程度，该连接所需的角位移要用逆运动学的方法来测定。仅为了展示这种方法同时减少方程的复杂程度，连杆的重力作用和惯性将忽略不计。关键词：正向运动学 逆运动学 自动化 封口机1简介一个自动化的工厂通常使用大量的电子控制的机械来完成生产，工厂的自动化对于其管理具有非常多的战略意义上的好处。标准的机械连接通常由电机、气动系统或电磁阀组成。在手动操作的机器上，人们通过目视来检测，其它方面的检测则是机械化地重复。这项研究的重点集中在一个品牌生产线中假想的用于压制和签名的封口机器，逆运动学的分析方法将使我们确定链接的角位移。运动学仅考虑物体的运动而忽略驱动力，而通过逆运动学的方法，则可以确定达到在最终效应下的指定目标所需要的链接角度和端部执行器的位置。Nagchaudhuri进行了一个关于曲柄滑块机构用于PID控制器使用的可行性研究，然而其中却忽视了偏移量。Tolani通过审查将逆运动学的求解问题分为七类，该些方法分别是牛顿迭代方法及其变体，还有雅可比矩阵和与伪逆的变体（也称为穆尔彭罗斯逆）方或非方形矩阵，以及其它基于控制理论和优化技术的方法。一些研究者提出了一些求解IK问题的算法，但不包括神经网络算法，循环坐标下降停止算法和不精确的策略算法，但像所有其他技术一样，对于一个给定问题的方法选择取决于问题的细节。Buss讨论了雅可比矩阵的转置，穆尔彭罗斯和阻尼最小二乘法，雅克比转置在计算量上较小然而在基于机器人的结构前提上却表现欠佳，在这项研究上，雅克比矩阵转置表现不如人意，但是其逆方法确实更加合适的，因为它将这个问题简化成了简单二维平面上四个自由度的问题。2 自动联接部分图1展示了机械密封系统的原理图。加盖或冲压由活塞或压装头实现，P.C是输送线，盖子或商标由红外线传感器感知并放置在正确的位置，S.依靠反馈传感器来密封或贴印商标。如果要印章、封盖或封装的物品离开了例子当中压装头将要接触的位置，传感器将反馈并使压装头缩回。有一系列的反馈给了电机控制器的话，它也可以不走得那么远。M.如果它是手动的，这种控制系统就类似于一个人工操作员,使用传感器和快速响应电动机控制器将使这个假设的机器成为一个在工厂中执行平常任务非常有用的工具。这个工厂支线由一个简单的曲柄滑块机构与致动器臂A.组成。用更清楚的话来说，如果盖子和容器在一条线上，那么指令将会发出使得活塞压下来完成密封；如果堵塞则扭转活塞；如果容器或盖子不在活塞头将不会压下；如果由磨损或撕裂所造成的密封长度短于预期长度则会加深压装力度。这清楚地表明，活塞确定链接的角度、方向和电机的运动。这就是一个逆运动学问题，传感器反馈部分是一个很复杂的控制工程问题，就不在此考虑了。 图1:自动封口机示意图（A：致动器臂 C：输送线 M：电机控制器 P：压装头 S：传感器 O：需要封口或封盖的物品）3 分析图2是一个代表性的曲柄滑块机构，从活塞轴到电机轴有一个偏移量f，O1、O2是轴的活塞移动的坐标(x,y)。电机相对O1顺时针或逆时针方向旋转，如果曲柄使位移s在活塞平面，它相当于一个ex和ey的运动。这个运动是由曲柄的顺时针或逆时针运动产生的，为连杆与曲柄之间夹角，也表示连杆与活塞平面之间的夹角。图2:偏移滑块曲柄(笛卡尔坐标系)在电脑的仿真中设计这些，角度会有明确的要求以防止连接不发生“物理分离”；对于一个真实物理连接环节，电机控制器需要控制移动的就只有曲柄。3.1 坐标系在这里采用笛卡尔坐标系，顺时针为正、向右移动为正、向上为正。上死点（TDC）的计算式如下，设曲柄半径为r,连杆长度为l,则有 其中，fm是指基于几何上的最大偏移量。下死点的计算如下上死点与下死点以及偏移量如图3所示。图3 上死点与下死点活塞被限制为只在平面方向移动，为向量，在这个研究中，方向向量上有，使其平面在45角至水平方向。3.2 正运动学电机顺时针方向移动产生的位移的位置以图2字母表示反映在（1）式中。在下标中以（i，f）分别表示起始和最终值，f的位置是在现实中平稳转动曲柄达到的，平滑度可以用数值方法细致渐进地达到要求。在这个增量的最后，最终的位移目标以一个角度参数的函数方式给出：角度的线性关系在这个问题中可以帮助减少在方程（1）计算中的自由度数量，由可以得出。利用三角法，可以计算出图2中任何时刻的活塞位置，见式（2）、（3）。由雅克比矩阵，给出了方程（4）同时将其简化为方程（5）。计算新的活塞位置涉及求解方程(1)，第一级下的泰勒级数下活塞的新坐标见方程（6）。是相关连接的位移矢量角，数学上，这里，我们设，由此活塞或压头的位置大约可由方程（6）得出。应该指出的是,可以取水平位置来进一步降低方程集，在这里可以将其称为0.3.3 逆运动学问题不在于给出Xi和来求解Xf，而是由给定Xi和Xf来求解。由迭代实得出了活塞的位移目标为，活塞的向量位移可表示为。因为这是一个没有其他方向位移的平面问题，所以可以将其简化为。为了减少可能的颠簸或跳动影响，可以逐步使用比基础上的r和L更直观的系数，前提是1且为逆雅克比矩阵。该算法可以检查是否已经达到目标，当求解少于预先确定的误差等级或达到最大数量迭代时，此迭代停止，这就是实时应用的一个关键部分。4 结果和讨论考虑活塞头在位置P时，机械臂在若干任意位置的方向，假设传感器系统需要活塞移动到新的位置P2，完成了几个任意起始位置的曲柄仿真所得出的结果都是同想要达到的目标接近的。如果曲柄当前方向与曲柄角之间为-5，且有一个指令从传感器发出使活塞压头收回其曲柄臂0.1倍的长度。模拟指示曲柄继续逆时针方向走15.58,这对应于0增加到19.26，相应的，减少到86.32。图4显示了模拟进展的活塞头从当前位置P1到新目标P2和完成迭代次数。图4:曲柄位置和迭代与雅可比矩阵求逆该技术使用的是逆雅克比技术，雅克比行列式的转置方法在这种情况下对于同样的问题是无法预测的，解决方法只有落定到一个角度内的局部最小值，但其收敛速度更快，见图5。图5 使用逆向和转置的雅克比矩阵得出的曲柄位置如果要求计算一个无法在实际上达到的目标，例如超过上死点或下死点位置的点P3，模拟运行并且在达到最大迭代数量时停止，亦可能是雅克比矩阵变为不可逆，见图6。图6 不可实现目标情况5 小结本文的重点是应用逆运动学技术分析机器人链接如获得封闭的自动化工厂，不考虑惯性效应的影响。反求技术的雅可比行列式,如上文献,在这个应用程序更可靠。雅可比行列式的方法是不可靠的转置。本文简要介绍了一个简单的应用了逆运动学的自动封口机；以活塞头收回0.1单位作为测试实例。新的曲柄角度可以由雅克比逆向方法相对雅克比行列式转置方法更精确地得出。这个问题还可以扩展到包括动力学对于最优转矩或电动马达驱动部件的可能性选择上。International Journal of Engineering and Technology Volume 2 No. 10, October, 2012An Inverse Kinematic Analysis of a Robotic SealerAkinola A. Adeniyi 1, Abubakar Mohammed 2, Aladeniyi Kehinde 31Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria2Department of Mechanical Engineering, Federal University of Technology, Minna, Nigeria3Department of Science Laboratory Technology, Rufus Giwa Polytechnic, Owo, NigeriaABSTRACTA 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, Sealer1. 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-Raphsons 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 schematic3. 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 s on the piston plane, it is equivalent to a motion of ex and ey. 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 centreThe 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). is 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 thatcan be measured from the horizontal to further reduce the equation sets, this is referred to as 0 elsewhere in this paper.3.3 Inverse Kinematics The problem is not that of solving for Xf given Xi andbut it is that of solving for given 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 ofwhich 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 of0 to 19.26and correspondingly,reduces 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 MatrixThe 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 MatrixIf 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 situation5. 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
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