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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 mail ncku edu 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 Compar d 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 cutt ng 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 Kinematicc d 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 l 1ain 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 x y z as shown in Fig lO c The relationship is given in the following 1 to 4 OA OB OC R O O O p 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 O p 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 z i 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 platform s 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 pointer s 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