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2016年第二届控制、自动化、和机器人的国际会议。基于三角形机器人运动学的球形运动地址在台湾省台南市大学路一号701，Chung-Ping Young和Yen-Bor Lin成功大学计算机科学与信息工程系，电子邮箱：cpyoung .tw.yen_bor .tw。摘 要工业中使用的机器人手臂被分为两类，包括串联机器人和并联机器人两类。与串联机器人相比，并联机器人具有精度高，刚度大、承载能力强、速度快、惯性小等优点。这次工作提出并实施了一种机制，基于Delta的机器的修改和实验，即可执行球形运动。实验结果表明，这种设计在使用中是实用和稳定的。根据定量评估,误差在几毫米之内。关键词：并联机器人 球面运动 Delta机器 自由度 逆运动学 机器人手臂。一 介绍工业中使用的机器人手臂被分为两类：包括串联机器人手和并联机器人手。如图一（a）和（b）所示， (a)串联机器人 (b)并联机器人图1.两类机器人工业手臂与串联机器人相比，并联机器人具有精度高，刚度大、承载能力强、速度快、惯性小等优点。Stewart平台1诞生之后其应用程序也诞生了，研究人员创造了许多不同的机制。Delta机器人是最受欢迎的解决方案之一，被广泛应用于许多领域。用户可以选择适合他们自由度和末端执行器来应用，如机器人手臂移动重物、3D打印喷漆、表面检查、如隐形眼镜质量检测、表面处理、和激光切割等。如今，基于笛卡尔式和三角形的机器是3D打印机中流行的两种类型。典型地，笛卡尔3D打印机放置一个方形的平台，它的头部运动被分解为x、y和z轴，且轴的每个方向的运动都由电机单独驱动。相比之下Delta 3D打印机将三个手臂布置成三角形结构，负载分部在三个部分，每个电机承受较小的负载。这种不同有利于打印的速度和准确性。虽然还有其他类型3D打印，那些都超出了本文的范围。末端执行器可以直接与对象交互，像喷涂绘画、挤压机、机械爬行、激光切割机、作为符合应用最终效应器。然而, 大多数常用的Delta机器人末端执行器仅限于平行移动到基础平台。虽然有几个机器人执行球形运动，但是他们都没有为Delta机器人专门设计。一般来说，需要更多的电机和更复杂的接头来使末端执行器完成球形运动并增加运动的自由度。在本文中，我们的目标是提出一个新的机制，以稍微增加或同等的成本使机器人执行球形运动。如移动平台和连杆的重新设计使得相应的逆运动学分析也可以实现这一任务，最初的Delta 3D打印机由三角形移动平台两侧的一对平行杆组成，如图2（a）所示， （a）平行联动 （b）拟议链接图2. Delta机器人框架的原始设计和建议设计他们确保移动平台的平面保持平行于基座，这种特性与我们执行球形运动的目标相冲突。为了支持球形运动，机器人需要使末端执行器偏移，俯仰和翻转为此还提出了一些辅助结构来使系统稳定，为此对固件进行了修改，以配合新提出的物理机制。为了导出末端执行器的方向和位置，正向运动学理论上是关节角度和连杆长度已知或测量时的方法。相反地，反向运动学是指定末端效应器的期望位置时导出关节角度的方法。由于前向运动学方法可能会遇到多种解决方案，因此逆向运动学将在这项工作中得以实现。有许多开源固件可以驱动包含Sprinter，RepRap，Grbl和Marlin的3D打印机，它们可以驱动Sprinter和Grbl。 用户可以修改配置以满足机器的需求。Marlin被选择在本工作中进行修改，以便为示范的实用性和性能提供示例实施。二 相关工作多项研究致力于机器人的球面运动。桂林杨采用三个相同的转动连接棱柱关节和球形（RPRS）腿来支撑移动平台2，Yan-Jin提出了一种选择性致动的并联机构，机器人的末端执行器可以执行6种自由度运动，即3自由度球面运动和3自由度平移三自由度运动3。虽然它们的机构设计与我们提出的机构设计不同，但运动学分析对于我们构建球形运动系统是有用的。1965年，斯图尔特发明了斯图尔特平台作为飞行模拟器。传统的斯图尔特平台使用六条可伸展腿。这是执行球形运动的最完整的并联机器人，其运动算法有助于设计我们的系统。Indrawanto介绍了Stewart平台的设计和控制，以讨论其特性和局限性。实验结果进行评估控制器的性能4。Mamoon提出了一种改进的Stewart平台，并允许使用便宜的步进电机作为执行器5。其他类似Delta或Stewart-lie的机器人也被开发出来。Patane.F开发了一种电动并联机器人，由一个由三个固定式线性电动执行器控制的移动底座组成，该执行器连接到相应的浮动和长度固定臂6。在文献7中，Xianqiang Y.使他们的机器人模仿人体肩部的运动，四根电缆在运动平台上对称分布。 电缆的一端连接到移动平台，另一端连接到地下室的电机。张力传感器和滑轮用于控制电缆。Angelm L.提出并行Delta型工业机器人的设计和硬件。他们还讨论了转向运动学特性和逆运动学8。 在9中，Aleksandrovich描述了一种新的三自由度操纵器。 机器人使用三条运动链，每条链包含一个平行四边形或两个位于底部的万向节。平四边形通过旋转对连接到基座。总之，上述机器人在机构和运动算法上是复杂的。 本文提出了一种简单的解析解法来简化球面运动的设计。三 实施本节介绍基于Rostock 3D打印机的示例实施，以表明所提出的设计能够成功实现目标。硬件组件和软件组件之间的关系如图3所示，Software LayerStep motorUkimaker1.5.7MarlinArduino mega 2560Hardware Layer图3.系统概述Arduino mega 2560被用来开发用于所提议的想法的程序。我们使用Ultimaker 1.5.7 pcb和A4988芯片来控制步进器和接收归位信号。不仅修改了硬件部分，还修改了软件部分以使它们正确地一起工作。 细节将在下面进行描述。A.硬件实现图4说明了这个示例实现的硬件体系结构。ArduinoMega2560PC SD cardUltimaker1.5.7pcbX-axis Homing SwitchY-axis Homing SwitchA4988Z-axis Homing SwitchA4988A4988 StepmotorY-axisStepmotorZ-axisStepmotorX-axis图4.硬件体系结构g代码是从连接的个人计算机或SD读卡器的串行端口获得的，收到的信息在Arduino mage 2560上进行分析和处理，然后发送给Ultimaker，通过GPIO信号达到1.5.7 pcb，pcb上的三个A4988芯片有助于发送控制信号来驱动步进器。此外，还有三个归位开关提供信号脉冲来终止归位过程，而末端执行器重新到达目标位置。Arduino mega 2560是基于ATmega2560微控制器的主板。 它与Ultimaker 1.5.7兼容，且开发环境良好。该Ultimaker也是一个电路板，并能够支持多达5个步进器。在我们提出的设计中只需要3个步进器。它采用高于12伏的电压来驱动步进电机，以获得更大的扭矩和更高的最大速度。其上的A4988芯片是全功能双极微步进电机驱动器，内置翻译器，处于停止，四分之一，八分之一以及十六分之一阶段模式。使用这些芯片，步进器的控制变得更容易，更多的引脚可以执行其他任务。罗斯托克3D打印机是由Johann在美国西雅图于2012年建造的线性三角洲3D打印机。Github和Thingiverse网站上发布了大量固件和相关开发工具。任何人都可以免费修改配置和软件包以适合指定的机器。如图5所示的原始图像是为这个示例实现而构建的。图5. 罗斯托克3D打印机最初如图6所示从左至右移除平行连杆。图6.去除并行链接对的一个链接因此，移动平台能够旋转（偏航，俯仰和滚转），但它变得不稳定并且失去了重复性。这意味着对于给定的执行机构位置不固定。为了保持稳定，需要考虑一些限制条件。如图2（b）所示，增加了三对弹簧。对称弹簧提供平衡力并防止移动平台偏航。在三对弹簧的作用下，移动平台再次变得稳定和可重复，并且还设计了一种新的控制该机器的算法。B.软件实现Marlin是选择的开源项目，它结合了名为Sprinter和Grbl的两个固件。它设计用于驱动控制面板，读取g代码，控制步进电机，控制挤出机，控制加热器以及操作SD卡。软件结构体系如图7所示，Use ProcessMain processMain libraryPlan motion libraryHardware Abstract LayerHardware DriveMessage receive/transmiterStepper libServo libHeater libLCD libSD librarySerial library图7.软件体系结构我们专注于两部分，包括运动算法和计划运动库。对于本文中的陈述，如图8所示的笛卡尔坐标用于声明符号，相对于x轴，y轴和z轴方向的旋转定义为滚动，俯仰和偏航。图8.Cartesian坐标此外，角度分别为a，b，和。对于拟议的联动机制，提出一项新动议模型是为了计算逆运动学而建立的。由于弹性平衡，禁止偏航。根据提出的设计，移动平台保持朝向基座的中心，如图9所示，图9.移动平台保持朝向基地的中心我们假设在工作平面上有一个虚拟中心标记为C，然后观察到移动平台上中心的轨迹是半径R到虚拟中心C的球体的一部分，如图10（a）所示。如图10（b）所示，从左到右：（a）移动平台上中心的轨迹（b）移动平台的圆柱形工作空间（c）三角机器人的运动学模型（d）位置矢量图 图10.球形轨迹和几何参数当移动平台沿着z轴方向移动时，该移动平台在工作空间内以顶部和底部的一部分球体在圆柱形状中移动。所提出的系统的几何参数将基于图10（c）中的符号和图10（d）中定义的位置矢量来导出。P1，P2和P3是移动平台的三个峰值。三个标记为T1，T2和T3的接头将移动平台连接到基座。因此，如图10（c）所示，存在两个协调系统：名为K（O-xyz）的固定全局协调系统和名为K（O-xyz）的局部协调系统。关系在下面的（1）至（4）中给出。OA=OB=OC=ROP1=OP2=OP3=r点A，B和C的坐标从以下（2）中获得。A=Rcos6 Rsin-6 ZB=Rcos2 Rsin2 ZC=Rcos-76 Rsin-76 Z类似地，从（3）获得点P1，P2和P3的坐标。P1=rcos6 rsin-6 0P2=rcos2 rsin2 0P3=rcos-76= rsin-76 0为了组合这两个坐标，分析位置矢量。如图11所示，图11移动平台从点M移动到点T.在从点M移动到点T的情况下。OP1=OO+OPi，i=1,2,3OT=OM+PmOP1=OT+Pt其中OM和OT是从O到O的向量，P，和P是从位置M到T和从位置T到P1的向量，其中t=1,2,3。由于矩阵计算是关联性的而不是交换性的，因此确定旋转矩阵的操作的排序是非常重要的。该顺序表示根据哪个方向旋转,请注意，提出的机器人不会偏航。我们的目标是在给出标记0的末端执行器的位置时计算三个执行器的坐标。如（5）所述，这些是T1，T2和T3，它们支配飞机Ti。Ti=TiX Tiy Tiz，i=1,2,3L2=(xi-Tix)2+(yi-Tiy)2+(Zi-Tiz)2，i=1,2,3其中L表示三个联系的公共长度。 然后导出下面的等式Tiz=L2-(xi-Tix)2-(yi-Tiy)2+Zi，i=1,2,3在此示例实施中，第一项的符号选择为负数。Marline软件包中名为calculate delta的函数计算执行器与目标坐标的位置，主要针对新提出的逆运动学修改输入。设置了一些基本配置来驱动使用过的主板以及步进器，并添加了名为DELTA FIXMID OFFSET的参数来表示移动平台与虚拟中心C之间的距离。4 实验结果检查建议设计的正确性，并按本节所述评估准确性。首先，在MATLAB中实现一个逻辑模型，以便可视化地观察状态。其次，进行了验证的物理实施，以证明其实用性和稳定性。最后，给出了数值评估以显示许多突发运行的性能。他们将在下面的小结中描述。A. 如图12所示，图12.逻辑模型以可视化方式呈现统计数据坐标在文本字段中给出，然后导出旋转角度并显示在以下两个字段中。有了这个工具，移动手势就清晰地展现出来了。B. 物理验证为了验证移动平台有意朝向虚拟中心，实施的3D打印机制作了一个半径5厘米，高7厘米，半球形的圆柱体，如图13左侧所示，图13.圆柱体和产生的半球。该特性已经过验证，使激光笔处于法线方向。 图14（a）表示当建议的3D打印机在固定高度移动并执行球形运动时激光点穿过同心图的轨迹，从左到右：（a）产生圆柱体和半球的移动路径 （b）侧面图14（a）。图14.保持向虚拟中心的检查。检查了不同的旋转角度0，并在IV-C部分给出了正确性和稳定性。开始时，激光点被校准为与底座垂直。 激光指示器在末端执行器上放置的任何轻微角度误差都会产生大量的位置误差放大距离效应。因此，我们用图6（a）所示的六个螺丝拧紧激光指示器，它们便于调节角度和位置。 我们焚烧移动平台，从工作区域的上限到下限遍历，并多次返回原位。如果激光点停留在半径为0.25mm的内圆区域，整个建议的系统应该是正确的，如图15所示，15（b）经过十次试验后。从这个实验结果来看，它已经完成了。从左到右（a）用六个螺丝调整位置和角度（b）激光指向有界区域图15.原始位置停留在内部圆形区域C. 定量评估安装了5V激光指示器作为末端执行器，并雇佣了一台附加的网络摄像机记录实验。定量评估以5个不同的旋转度进行，测试程序使用OpenCV库进行编码，以便在机器执行测试项目时记录视频数据。 我们计算了从激光指示器位置到工作平面中心位置的位置偏移量，正如第IV-B节所述的10次爆发错误。 在工作平面中心位置坐标为（307,183）的情况下，位置偏移形式的位置误差如表1所示。每个像素代表0.192mm，通过测量的13个像素的0.25cm距离得出。总之，如图16所示，图16.10次爆炸后旋转角度不同的位置误差随着旋转度增加，位置偏移从0.7mm增加到2.91mm。错误来自不平衡的弹性和关节摩擦。这表明激光指向工作平面的轨迹组装在中心位置的狭窄区域。因此，这种设计是稳定的，准确性是可以接受的。我们还将我们提出的系统的生产性能与工业市场上的5轴数控机床的生产性能进行了比较，如表II所示。与具有相似的执行球形运动能力的类似机器相比，发现成本显着较低。记录的激光位置旋转角度5.0010.0015.0020.0025.00X位置平均313.00310.67306.26301.49293.01Y位置平均185.75186.41187.52184.73180.68最大错误1.873.563.947.1611.96X坐标的最大误差1.352.883.136.449.42Y坐标的最大误差1.291.922.413.137.37产生性能的比较机器自由度准确性工作区价格CROSS-I I06ill,30.005900x600x6001500000AweaFV-96050.01960x600x4801200000CNC 3040 Table ColumnType Engraving Machine50.02300x400x15050840This Work3470x70x200110005 结论和未来的工作所提出的系统在Arduino mega 2560平台上引入并实现，以获得实验结果，功能得以实现，位置偏移得到了良好的控制。该系统是一个很好的解决方案，可以满足几毫米精度的应用要求。 未来，电子弹簧被认为是集成在一起，以更精确的方式进行控制，以提高该系统的精度。参考文献：1 B. Dasgupta and T. Mruthyunjaya, T he stewart platform manipulator:a review, Mechanism and Machine Theory, vol. 35, no. I, pp. 15-40,2000.2 G. Yang, I.-M. Chen, W. Chen, and W. Lin, Kinematic design of asix-dof parallel-kinematics machine with decoupled-motionarchitecture, Robotics, IEEE Transactions on, vol. 20, no. 5, pp. 876-887, Oct 2004.3 Y. Jin, I.-M. Chen, and G. Yang, Kinematics analysis of a 6-dofselectively actuated parallel manipulator, in Robotics, Automationand Mechatronics, 2004 IEEE Conference on, vol. I, Dec 2004, pp.231-236 vol.l. 4 Indrawanto and A. Santoso, Design and control of the stewartplatform robot, Asia International Conference on Modelling &Simulation, vol. 0, pp. 475-480, 2009.5 M. Mamoon and Saifullah, Inverse kinematics and path planning ofstewart platform using crank arm actuation system, in AppliedSciences and Technology (IBCAST), 2014 11th InternationalBhurban Conference on, Jan 2014, pp. 175-18.6 F. Patane and P. Cappa, A 3-dofparallel robot with spherical motionfor the rehabilitation and evaluation of balance performance, Neura lSystems and Rehabilitation Engineering, IEEE Transactions on, vol.19, no. 2, pp. 157-166, April 2011.7 X. You, W. Chen, S. Yu, and X. Wu, Dynamic control of a 3-dofcabledriven robot based on backstepping technique, in Industr ialElectronics and Applications (ICIEA), 2011 6th IEEE Conference on,June 2011, pp. 1302-1307.8 L. Angel, J. Bermudez, and O. Munoz, Dynamic optimization andbuilding of a parallel delta-type robot, in Robotics and Biomimetics(ROBIO), 2013 IEEE International Conference on, Dec 2013, pp.444-449.9 M. Aleksandrovich, S. Sergeevna, and M. Yurievich, Determinationof motion freedom and direct kinematic problem solution of themechanism similar to delta robot, in Electrical Engineering,Computing Science and Automatic Control (CCE), 2014 11thInternational Co nf erence on, Sept 2014, pp. 1-5.2016 The 2nd International Conference on Control, Automation and Robotics The Spherical Motion Based on the Inverse Kinematics for ADelta Robot Chung-Ping Young and Yen-Bor Lin Dept of Computer Science and Information Engineering National Cheng-Kung University No.1, University Road, Tainan City 701, Taiwan e-mail: cpyoung.tw.yen_bor.tw Abstract-The robot arms used in industry are classified as two categories including the se rial robot and the parallel robot. Compared with the serial robots, parallel robots have advantages in high precision, high stiffness, high load, high speed and low inertia. This work proposed and implemented a mechanism to perform spherical motion with merely extra modification of a Delta-based machine and the experimental results showed that this design is practical and stable for usage. According to the quantitative evaluation, the errors are within several millimeters. Keywords-Parallel Robot; Spherical Motion; Delta Machine; Degree 0/ Freedom; Inverse Kinematics; Robot Arm; 3D printing I. INTRODUCTlON The robot arms used in industry are c1assified as two categories inc1uding the serial robot and the parallel robot. They are illustrated in Fig. l(a) and Fig. l(b). Compard with the serial robots, the parallel robots have advantages m high precision, high stiffness, high load, high speed and low inertia. Researchers created lots of mechanisms for different applications after the Stewart platform 1 was born. The Delta robot is one ofthe most popular solutions and has been widely used in many domains. Users choose the degree of freedom (DOF) and the end-effector to fit their applications such as the robot arms to move heavy objects, the 3D printer, the painting, the surface inspection such as the contact lenses quality detection, the surface processing and the laser cuttng. Nowadays, the Cartesian-based and the Delta-hke machines are the two main types of popular 3D printers. Typically a Cartesian 3D printer places a squared bed, and the movement of its head is decomposed to the directions of the X, Y and Z axes separately. The motion in each direction ofthe axes is driven by a motor individually. By contrast, the Delta 3D printer arranges the three arms in a triangular configuration and the load is distributed to three parts so each motor bears less. The difference benefits the printing speed and the accuracy. Although there are still other types of 3D prints, those are out of the scope in this artic1e. The end-effector interacts with the object directly. A spray painter, an extruder, a mechanic craw, a laser cutter, a pen. or a drill is frequently chosen to be the end-effector accordmg to the application. However, the end-effector of most commonly used Delta robots is restricted to move parallel to the base platform. Although there have been several robot mechanisms to perform the spherical motion, none ofthem is designed for a Delta robot. Generally, more motors and more complicated joints are needed to make the end-effector perform the spherical motion and increase the DOF of movement. In this paper, the goal is to propose a new mechanism with slight increased or similar cost to make the Delta robot perform the spherical motion. The moving platform and the link sticks are redesigned such that the corresponding inverse kinematics analysis is also given to achieve this mission. ._,/A Kinematiccd effector Fixedbase m Endeffector K inematicchai Fixedbase (a) The serial robot. (b) The parallel robot. Figure 1. The two categories of industrial robot arms. (a) The parallelIinkage. (b) The proposed linkage; Figure 2. The original and proposed designs ofDelta robot hnkages. The original Delta 3 D printer consists of a pair of parallel sticks on each side of the triangle moving platform as shown in Fig. 2(a). They ensure that the plane of the moving platform stays parallel to the base. This characteristic conflicts with our goal to perform the spherical motion. To support the spherical motion, the robot is required to make the end-effector to yaw, pitch and roll. Certain assistant structure is also proposed to make the system stable. The firmware is modified to cooperate with the new proposed physical mechanism. To derive the orientation and the position of the end-effector, forward kinematics is theoretically the method when the joint angles and the link lengths are known or measured. Oppositely, the inverse kinematics is the method to derive the joint angles when the desired position of the end-effector is specified. The inverse kinematics will be derived in this work because the forward kinematics method may suffer multiple solutions. There are many open source firmwares to drive the 3D printers inc1uding Sprinter, RepRap, Grbl, and Marlin whi.ch combines Sprinter and Grbl. It is possible for users to modlfy the configuration to meet the demand of the machine. Marlin 978-1-4673-9859-6116/$31.00 2016 IEEE 29 is chosen to modify in this work to have the sampie implementation for the demonstration of the practicability and the performance. 11. RELATED WORK Several research es have devoted to the spherical motion ofrobot arms. Guilin Yang employed three identical revolute joint, prismatic joint and spherical joint (RPRS) legs to support the moving platform 2. Yan-Jin presented a selectively actuated parallel mechanism and the end-effector of the manipulator is able to perform 6 DOF motion, those are 3 DOF spherical motion and 3 DOF translation with three limbs 3. Although their mechanism designs are different from that of our proposed mechanism, the kinematics analysis is useful for us to build a spherical motion system. In 1965, Stewart invented the Stewart platform as a flight emulator. The conventional Stewart platform uses six extensible legs. This is the most completed parallel robot to perform the spherical motion and its motion algorithm is helpful to design our system. Indrawanto presented the design and the control of aStewart platform to discuss its characteristic and the limitation. The experimental results were conducted to evaluate the performance of the controllers 4. Mamoon proposed a modified Stewart platform and the cheap stepper motors were allowed to be used as the actuators 5. Other Delta-like or Stewart-lie robots were also developed. Patane, F. developed an electrically actuated parallel robot consisting of a moving base controlled by three stationary linearly electrical actuators connected to the corresponding floating and length-fixed arms 6. In 7, Xianqiang Y. made their robot imitate the motion of human shoulder and four cables were symmetrically distributed in the moving platform. One endpoint of the cable is connected to the moving platform and the other is connected to the motor in the basement. The tension sensor and the sliding wheel were employed to control the cables. Angelm L. proposed the design and the hardware of a parallel Delta-type industrial robot. They also discussed the properties of the forwarding kinematics and the inverse kinematics 8. In 9, Aleksandrovich described a new manipulator with three DOF. The robot uses three kinematic chains and each of them consists of a parallelogram or two universal joints located at the base. The parallelograms were connected to the base by the rotating pairs. In summary, the above-mentioned robots are complicated in the mechanism and the motion algorithrn. A simple and easier solution is proposed in this paper to simplify the design ofperforming the spherical motion. III. IMPLEMENTAnON This section describes the sampie implementation based on the Rostock 3D printer to show that the proposed design achieves the goal successfully. The relationship between the hardware components and the software components is illustrated in Fig. 3 Arduino mega 2560 was used to develop the program for the proposed idea. We used Ultimaker 1.5.7 pcb with A4988 chips on it to control the steppers and to 30 receive the homing signal. Not only the hardware part but also the software part was modified to make them work together correctly. The detail will be described in the folIows. Figure 3. The system overview. Figure 4. The hardware architecture. A. Hardware Jmplementation Fig. 4 illustrates the hardware architecture of this sampie implementation. The g-code was obtained from the serial port of connected personal computer or the SD card reader. The incoming information was analyzed and processed on Arduino mage 2560, and then was sent to the Ultimaker 1.5.7 pcb via GPIO signals. The three A4988 chips on the pcb help to send control signal to drive the steppers. Besides, there are three homing switches pro vi ding the signal pulse to terminate the homing process while the end-effector reaches the target position. The Arduino mega 2560 is a board based on the A Tmega2560 micro-controller. It is compatible with Ultimaker 1.5.7 and the development environment is well-provided. The Ultimaker is also a board and is able to support up to 5 steppers. Only 3 steppers are required in our proposed design. It was employed at higher voltage than 12 voltage to drive the stepper motors with more torque and higher maximum speeds. The A4988 chips on it are fully-featured bipolar microstepping motor drivers with built-in translator in full, halt, quarter, eighth and sixteenth-step modes. With these chips, the control over the steppers becomes easier and more pins are leisure to perform other tasks. The Rostock 3D printer is a linear delta 3D printer built in 2012 by Johann in Seattle, USA. Lots of firmware and related development tools were published on the Github and Thingiverse websites. It is free for anybody to modify the configuration and the software packages to fit a specified machine. The original one as shown in Fig. 5 was built for this sampie implementation. The parallellinkages were removed initially as shown in Fig. 6 from left to right. As result, the moving platform was able to rotate (yaw, pitch, and roll) but it became unstable and lost the repeatability. It means the position of the endeffector is not fixed for a given wished actuator position. In order to make it stable, some restriction is necessary to be considered on it. As shown in Fig. 2(b), three pairs of springs are added. The symmetrie springs offer the balance force and prevent the moving platform yawing. With the involvement of the three pairs of springs, the moving platform becomes stable and repeatable again, and a new algorithm to control this machine was also designed as the following statements. =?ff Figure 6. The Removal of one Iinkages of parallel linkage pairs. User Process l1ain proccss I Mein l ihmry I I Plan motion library I .-:-:-_-,-.,.-.,H _dw-,are Abst,-,.ct _L.Ve.,.,.,.-:.,-_-, Mes.5aJte reeeiver/transmiter Hardware: Driver 11 Serial libml) 1 1 SO l ibml) 11 I Stepper Jih. I I Servo lib. 1 LCD lib. 1 1 He.edib. Figure 7. The software architecture. Roll y Figure 8. T he Cartesian coordinate. B. Software lmplementation Marlin is the chosen open source project which combines the two fIrmwares named Sprinter and Grbl. It was designed to drive the control panel, to read the g-code, to control the step motors, to control the extruder, to control the heater, and to operate the SD card. The software architecture is iIIustrated in Fig. 7. We focus on the two parts including the motion algorithm and the plan motion library. For the statements in this paper, the Cartesian coordinate as shown in Fig. 8 is used to declare the notation. The rotations with respect to the x-axis, y-axis and z-axis directions are defmed as roll, pitch and yaw. Besides, the angles are a, and y respectively. For the proposed linkage mechanism, a new motion model is required to build in order to compute the inverse kinematics. The yaw is forbidden due to the elasticity balance. According to the proposed design, the moving platform keeps toward to the central of the base as shown in Fig. 9. We assume that there is a virtual center notated as C on the working plane. Then, it is observed that the trajectory of the center on the moving platfonn is a part of the sphere with radius R to the virtual center C as shown in Fig. IO(a). The moving platfonn moves in the workspace in the shape of a cylinder with apart of sphere on the top and the bottom when it moves along the z-axis direction as shown in Fig. lO(b). The geometrie parameters of the proposed system will be derived based on the notation in Fig. 10(c) and the position vectors defmed in Fig. lO(d). PI P2 and P3 are the three peaks of the moving platfonn. Three joints labelIed T T2 and T3 link the moving platform to the base. Therefore, there are two coordination systems: a fIxed global coordination system named K (O-xyz) and a local coordination system named K (O-xyz) as shown in Fig. lO(c). The relationship is given in the following (1) to (4). -OA=OB=OC=R O =O =Op:, =r (1) The coordinates of the points A, Band C are obtained from the following (2). A = Rcos() Rsin(-t) z B= Rcos(t) Rsin(t) z C=RCOS(_7;) Rsin(_7;) z (2) Similarly, the coordinates ofthe points PI, P2 and P3 are obtained from (3). = cos() rsin-t) 0 = COS(I) rsin(I) 0 J;=COS-7;) rsin-7;) 0 (3) To combine the two coordinates, the position vectors are analyzed. In the case of moving from point M to point T as shown in Fig. 11, Op, = OO+Op,i = 1,2,3 OT=OM+, OI. =OT+p (4) where OM and OT are the vectors from 0 to 0, P,n and p, are the vectors from position M to T and from position T to Pt where t=I,2,3. 31 Figure 9. The moving platform keeps toward to the central ofthe base. t. jll i ! . , , I . . From left to right: (a) The trajectory ofthe center on the moving platform. (b) The cylindrical workspace ofthe moving platform (c) The Kinematics model ofa delta robot. (d) The position vectors. Figure 10. The spherical trajectory and the geometrie parameters. IL-_ -T , Figure 11. T he moving platform move from point M to point T. It is essential to determine the ordering of the operations of the rotation matrices because matrix calculation is associative but not commutative. The ordering presents which direction the rotation according to. Note that the proposed robot does not yaw. The goal is to calculate the coordinates of the three actuators when the position of the end-effector labelled as 0 is given. These are TI. T2 and T3 these dominate the plane Ti as described in (5). T, = T,x T,y T,z, i = 1,2,3 (5) L2 = (x, -T,J2 + (y, -T,y)2 + (z; -T,z)2,i = 1,2,3 (6) where L denotes the common length of the three linkages. Then the following equation is derived: T,z =JL2 -(x, _T,J2 -(y, _T,y)2 +zi = 1,2,3(7) where the sign of the fIrst term is chosen to be negative in this sampie implementation. The function named calculate delta in the Marline package calculates the position of the actuators with the target coordinate as input was modifIed mainly for the newly proposed inverse kinematics. Some basic confIgurations were set to drive the used motherboard as weH as the the steppers, and the parameter named DEL TA FIXMID OFF SET was added to represent the distance between the moving platform and the virtual center C. 32 Figure 12. The logical model to demonstrate the status visually. Figure 13. The cylinder and the produced hemisphere. From left to right: (a) The moving path to produce the cylinder and a hemisphere on it. (b) The side view ofFig.14(a). Figure 14. T he examination ofkeeping toward to the virtual center. IV. EXPERIMENT AL RESUL T The correctness of the proposed design was examined and the accuracy was evaluated as described in this section. First, a logical model was implemented in MA TLAB to observe the status visually. Secondly, the physical implementation for the verifIcation was performed to prove the practicability and the stability. Finally, the numeric evaluation was given to show the performance with many burst runs. They will be described in the following subsections. A. Visual Simulation As shown in Fig. 12, the co ordinate is given in the text fIelds and then the degrees of rotation are derived and shown in the following two fIelds. With this tool, the moving gesture is c1early demonstrated. B. Physical Verification To verify the moving platforms keeping intentionally toward to the virtual center, the implemented 3D printer made a cylinder with 5 cm radius and 7 cm height and a hemisphere on it as shown in the left ofFig. 13. The property was verifIed, and this made the laser pointer be on the direction of the normalline. Fig. 14(a) illustrates the trajectory where the laser point crossed the concentric drawing when the proposed 3D printer moves at a fIxed height and performs the spherical motion. Different degrees of rotation e were examined and the correctness and the stability will be given in section IV-C. At the beginning, the laser point was calibrated to be perpendicular to the base exactly. Any slight angular error of the placement of the laser pointer on the end-effector produces a huge number of location errors due to the amplification from the distance effect. Therefore, we screwed the laser pointer with the six screws as shown in Fig. 15(a) and they make it easier to adjust the angle and the position. We burned the moving platform to traverse around the work space from the upper limit to the lower limit and back to the horne position several times. The whole proposed system is supposed to be correct if the laser point stays in the inner circular region with the radius of0.25 mm as shown in Fig. 15(b) after ten runs of the bumed tests. As results from this experiment, it was accomplished. /. . , ei , From left to right: (a) The six screws to tune the position and the angle. (b) The laser points to the bounded region. Figure 15. The horne position stays in the inner circular region. 18.00 . ,s 6.00 DofVU Figure 16. The position error after 10 burst runs with different rotation degrees. C. Quantitative Evaluation A 5V laser pointer was installed as the end-effector and an additional web camera was hired to record the experiments. The quantitative evaluation was performed with 5 different rotation degrees. The testing program was coded with the OpenCV Iibrary to record the video data when the machine performed the testing items. We ca1culated the position offsets from the laser pointers position to the central position of the working plane as the errors for 10 burst runs as described in section IV -B. The errors of the position in the form of positional offset are shown in Table I in the case that the coordinate of the central position of the working plane is at (307,183). Each pixel represents 0.192mm derived by the measured 0.25cm distance for 13 pixels. In summary, as shown in Fig. 16, the position offset grew from 0.7 mm to 2.91 mm as the rotation degree increased. The errors come from the unbalanced elastic and the joint friction. This shows that the trajectory which the laser pointed to on the working plane is assembled in a narrow region of the central position. Thus, this design is stable and the accuracy is acceptable to practice. We also compared the productive properties of our proposed system to those of 5-axis CNC machines in the industrial market as shown in Table 11. It was found that the cost is significantly low as compared to the comparable machines