IROS2019国际学术会议论文集 1526

A Mobile Extendable Robot Arm Singularity Analysis and Design Seiichi Teshigawara and H Harry Asada Member IEEE1 Abstract Inspection and maintenance of equipment inside buildings such as exit signs bared pipelines air vents and fi re alarms often requires a robot to reach high hidden or confi ned areas that are diffi cult for humans to access Even though these tasks are easy and repeatable they are still not automated The Mobile Extendable Robot Arm MERA is a movable robot arm with a novel 2 DOF scissor mechanism for reaching a high place and positioning an end effector MERA is composed of a locomotion vehicle with a rotation table and a 4 DOF extender arm itself made of two layers of the 2 DOF scissor mechanism arranged in series Placing the end effector at an arbitrary point in space the 4 DOF arm possesses two degrees of redundancy allowing access to a point from various directions and enabling obstacle avoidance In this paper we present the design and analysis of the 2 DOF scissor mechanism The 2 DOF scissor mechanism has two rotary actuators for driving the base links individually consequently the mechanism can elongate the entire body and tilt at the center of the base shaft However we found that the 2 DOF scissor mechanism had a singularity after analyzing the singularity we propose two novel solutions to the problem I INTRODUCTION Inspection and maintenance workers working inside large buildings such as factories hospitals and schools are fre quently tasked to access narrow spaces For example older buildings often have many bared pipelines and air ducts especially on the ceilings of basements and these often stretch in various directions in a complex manner Workers can access these pipelines from below by using a ladder However because maintenance tasks like this can often result in worker injury and decreased productivity facility staff should not perform every required task manually Thus an effective technology is needed to automatically perform easily repeatable but high risk tasks such as the inspection of exit signs bared pipelines air vents fi re alarms and the insides of fl uorescent light covers in order to maintain the safety of users of these buildings One solution is QuicaBot 1 a quality inspection and assessment robot that can autonomously scan an entire room using cameras and laser scanners to pick up building defects such as hollowness and cracking and measure evenness alignment and inclination This technology is usable for inspection of fl oors walls and ceilings However it is diffi cult to use in confi ned areas and the detection accuracy relies on the resolution of the cameras and scanners Other solutions are drones 2 and wall climbing robots 3 4 They can easily access higher places but these robots This work has been supported by NSK Ltd 1 The authors are affi liated with the dArbeloff Laboratory for Information Systems and Technology in the Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge MA 02139 USA seiichi asada mit edu have a low payload capacity and unsafe for usage inside buildings Zipper type 5 6 and chain type 7 8 mech anisms possess high rigidity and a high rate of expansion However these extendable mechanisms extend only directly upward Therefore it is diffi cult to access confi ned areas with complex structure On the other hand TSE Triple Scissor Extender 9 TSERA Triple Scissor Extender Robot Arm 10 realized a high payload capacity and multiple degrees of freedom However because these robots are base mount type and have a singularity in their scissor mechanisms their workspace is limited by their geometric design To address this limitation we propose the Mobile Extend able Robot Arm MERA see Fig 1 MERA is composed of a locomotion vehicle with a rotating base and a 4 DOF extender arm itself made of two layers of 2 DOF scissor mechanisms arranged in series Each scissor mechanism layer has one revolute and one prismatic degree of freedom The fi rst layer is high power and long range on the other hand the second layer is small and short range in order to be able to access a confi ned area easily As the 4 DOF extender arm moves in one plane 2 degrees of freedom are redundant this allows access to a target point from above avoiding obstacles such as pipelines see Fig 2 Fig 1 The Mobile Extendable Robot Arm MERA has two layers of 2 DOF scissor mechanisms The fi rst layer is high power and long range and the second layer is small and short range so as to be able to access confi ned areas In this paper we present the design kinematics and singularity analysis of the 2 DOF scissor mechanism that is the key component of the MERA We found that the 2 DOF scissor mechanism has a singularity specifi cally it is impossible to elongate the mechanism directly upward After analyzing the singularity we propose a novel solution to the problem 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE5131 Fig 2 Inspection of a pipeline using MERA MERA is able to reach the upper side of the pipeline and access a confi ned space with its small extendable arm II PROTOTYPE OF THE2 DOFSCISSOR MECHANISM The main benefi t of using a scissor mechanism is its characteristically large expansion rate Typically scissor mechanisms are used as lifters that provide portability and accessibility However since the traditional scissor mecha nism can only extend in a direction perpendicular to its base the work space is limited The 2 DOF scissor mechanism has two rotary actuators for driving the base links individually the mechanism can both elongate the entire body and tilt at the center of the base shaft see Fig 3 Fig 3 Prototype for the 2 DOF scissor mechanism Actuation of the two different DOF prismatic and revolute is shown at right If the two base links rotate in opposite directions the mechanism extends or contracts If the base links rotate at the same speed in the same direction the mechanism rotates around the base shaft maintaining its length This driving method has a major advantage for torque and extensional speed of the robot arm When the arm rotates the two motors rotate the links to the same direction Thus the rotation torque of robot arm is double the torque of one motor On the other hand when the arm only expands the motors drive the links at equal speeds and opposite directions In this case the relative rotation speed between the linkages is double the speed of one motor As a result the arm can expand and contract very quickly This driving method is known as coupled drive 11 III KINEMATICS In this section we show the kinematics of the 2 DOF scissor mechanism A Forward kinematics First we will analyze the forward kinematics Forward kinematics refers to the computation of the position of the end effector from joint displacements A coordinate system is set around the rotation shaft of the base link as shown in Fig 4 The forward kinematics equations are shown below xn 1 2L cosq1 cosq2 1 yn 1 2L sinq1 sinq2 2 Here L is defi ned as L 2 li the total length of one series of links and the theoretical maximum extension length Fig 4 Cartesian coordinate model of an ideal n stage 2 DOF scissor mechanism B Inverse kinematics Next we will calculate the inverse kinematics Inverse kinematics makes use of the system geometry to determine the joint displacements that provide a desired position for each of the robots end effectors The inverse kinematics equations are shown below q1 atan2 yn xn cos 1 p x2 n y2n L 3 q2 atan2 yn xn cos 1 p x2 n y2n L 4 5132 IV MODELING FOR SINGULARITY ANALYSIS We implemented the 2 DOF scissor mechanism in a prototype see Fig 3 and attempted to extend it upward However only the bottom links extended and only slightly the middle and the upper links cannot be extended as if the mechanism were locked see Fig 5 We expected that this is due to a mechanical singularity for several reasons Primarily the angle of the links is close to 180 degrees the singular angle of an ideal scissor linkage Additionally we considered how structural non idealities specifi cally the defl ection of the each link and the clearance of each joint can contribute toward singular behavior Fig 5 Demonstration of an attempt to extend the initial 2 DOF scissor mechanism prototype It is diffi cult for the scissor to expand upward even when only loaded by self weight specifi cally upper segments of the scissor effectively lock even when lower segments move slightly In this section we detail the modeling of the 2 DOF scissor mechanism for singularity analysis There are many papers which analyze scissor mechanisms in order to opti mize their deployment as actuators 12 13 14 However these papers did not focus on singularity analysis because commercial 1 DOF scissor lifts avoid the singularity by supplementing the scissor mechanism with a high power linear actuator We based our model on that developed by Chikahiro et al 15 However their model did not account for clearance at the joints thus we modifi ed it to include this phenomenon and predict its effects Our simulation model can calculate the motor torque needed to drive the scissor mechanism the load at each joint the defl ection of the whole structure and the infl uence of the clearance of each joint A Calculation of the forces added each joint A free body diagram FBD for a single scissor segment is shown in Fig 6 the length of each member is l1 l2 and the inclination angle of the scissor is measured from the vertical direction This scissor structure was designed by using the equilibrium equations for the FBD The equilibrium equations for each external force in the x and y directions are as follows H FA1x FB1x FD1x FE1x 0 5 V FA1y FB1y FD1y FE1y 0 6 Fig 6 Free body diagram FBD for a single segment of the 2 DOF scissor mechanism For the intersecting members two equilibrium equations can be obtained for the moments at Point C MC BD and MC AB MC BD 1FB1x 1FB1y 2FD1x 2FD1y 0 7 MC AB 1FA1x 1FA1y 2FE1x 2FE1y 0 8 Where 1 l1cos 2 l2cos 1 l1sin and 2 l2sin First consider the case of a cantilever model that has pinned support at Points A1and B1 The matrix equation shown in 9 is directly constructed from the equilibrium equations 5 8 Equation 10 shows a block vector ab breviation of 9 1010 0101 1 100 00 1 1 FA1x FA1y FB1x FB1y 1010 0101 00 2 2 2 200 FD1x FD1y FE1x FE1y FC1x FC1y 0 0 9 L1 FA1FB1 T R2 FD1FE1 T FC10 T 10 Where L1 R2 R4x4 Next we consider the modeling of the entire 2 DOF scissor mechanism including the tip and base links see Fig 7 In this problem it is possible to treat the external forces as the left and right segments of internal forces operating on the hinges like at Points Dn 1Anand En 1Bn These relationships can be expressed as 11 Equation 12 shows a block vector abbreviation of 11 FDn 1x FDn 1y FEn 1x FEn 1y FDn 1Anx FDn 1Any FEn 1Bnx FEn 1Bny FAnx FAny FBnx FBny 11 FDn 1FEn 1 T FDn 1AnFEn 1Bn T FAnFBn T 12 5133 Fig 7 Model of the Entire 2 DOF Scissor Mechanism Moreover the section forces for each segments can be calculated in a manner similar to that for the single segment scissor problem These relationships can be expressed as 13 The equations 12 and 13 can be applied repeatedly to each segment from segment n 1 to segment 1 As the result we can obtain the load at each joint FAn 1FBn 1 T L 1 n 1Rn FDn 1FEn 1 T L 1 n 1 FCn 10 T 13 Fig 8 b shows the FBD of the tip links From this FBD we can derive the equilibrium equation as follows FAnFBn T L 1 n Rn FCn0 T 14 Fig 8 FBD for a Base links and b Tip links Represented as inputs Anand Bn the loads at the points D0and E0 can be obtained from 12 and 13 We show the FBD of the base links in Fig 9 a From this fi gure we can derive the equilibrium equation as follows in 15 Fox Foy 1 2 R1 FDox FDoy FEox FEoy Fo T R1 FDoFEo T 15 From 15 we can obtain the motor torque needed to drive the scissor mechanism When we perform the simulation the self weights of components are modeled as concentrated loads Specifi cally the self weights of links are assigned to points Di 1Aiand Ei 1Bi i 1 2 n and the self weights of joints are assigned to points Ci Di 1Aiand Ei 1Bi i 1 2 n The payloads external forces Wx Wy are assigned to point Cn represented as the force FCn B Calculation of the deformation of each link Next we show the defl ection modeling for each link using mechanics of materials The load at each joint can be obtained as per the above equation and calculation As a result we can estimate the defl ection of each link First we calculate the load perpendicular to the link Fig 11 shows one segment for clarifi cation and simplicity The load for each link PDn 1 PEn 1can be obtain as follows PDn 1 PEn 1 cos sin 00 00 cos sin FDn 1x FDn 1y FEn 1x FEn 1y 16 Fig 9 FBD for calculating the deformation of one segment of the linkage Each crossing link is modeled as an overhanging beam shown at right Next we selected the model of an overhanging beam as shown in Fig 9 The defl ection Dn 1 En 1 of the beam can be obtained as per 17 E N mm2 and I mm4 correspond to the Young s modulus and second moment of area This model holds for all segments except for the base link Instead we modeled the base link as a cantilevered beam see Fig 10 5134 The defl ection D0 E0 of the beam can be obtained as per 18 Dn 1 PDn 1l2 n ln 1 ln 3EI En 1 PEn 1l2 n ln 1 ln 3EI 17 D0 PD0l3 1 3EI E0 PE0l3 1 3EI 18 Fig 10 FBD for calculating the deformation of the base links Each crossing link is modeled as a cantilevered beam shown at right C Calculation of new joint positions As is mentioned above each link is defl ected by a self weight and external forces and which causes the defl ection of the whole scissor structure The deformation due to the link defl ection on the XY plane can be determined by 17 or 18 and the following equation Dn 1x Dn 1y En 1x En 1y cos 0 sin 0 0 cos 0 sin Dn 1 En 1 19 Moreover when we simulate the displacement of each joint position we must consider the clearance that contains various tolerances gaps and deformations in real mechanical joints For example Radial internal clearance of ball bearings This factor depends on the kind of ball bearings We can obtain this value from data sheets a standard miniature ball bearing used in our 2 DOF extendable arm has a clearance between 5 m and 10 m Deformation of ball bearings depending on an added load Though this factor is generally smaller than radial internal clearance it should be considered in the case of a large load The tolerance between the ball bearing and its housing The tolerance between the shaft and the ball bearing Ideally every clearance parameters must be estimated in the calculation however we simplify by setting one clearance parameter Ga and using this parameter to describe the clearance of every joint of the 2 DOF scissor mechanism Because this clearance holds for each joint there will be a displacement between the center positions of each joint between a back link and a forward link as shown in Fig 11 Furthermore the direction of the clearance Gadepends on the load direction at each joint The direction at Dn 1 and En 1can be obtained as per 20 By using the the clearance on the XY plane can be obtained using 21 and as mentioned above this parameter is applied to all joints Fig 11 The direct infl uence of a clearance Gaat a joint There arises a positional deviation of magnitude Gain the same direction as the joint contact force tan 1 Fy Fx 20 Gx Gacos Gy Gasin 21 The deformation of the entire 2 DOF scissor mechanism on the XY plane is determined by the defl ection of each link and the clearance of each mechanical joint Therefore the new position of each joint is calculated as follows D0 D Dx Dy tD Dx Gx Dy Gy 22 E0 E Ex Ey tE Ex Gx Ey Gy 23 Similarly the calculation is performed for all scissor segments and all points are redrawn from the base joint to next joint This is illustrated in Fig 12 Fig 12 Redrawing the joint positions of the linkage after link deformation and joint clearance deformation are taken into account 5135 V SINGULARITYANALYSIS In this section we show the singularity analysis of the 2 DOF scissor mechanism First we expect that the cause of singularity is the angle of base link 170 deg as it close to the angle of the theoretical singularity posture 180 deg To verify this we simulated the necessary motor torque to extend the fully contracted scissor mechanism The simulation parameters are shown as follows Material Aluminum 6063 T5 Density 2690 kg m3 Young s modulus 68600 MPa Cross section square pipe outer dimension w 20 mm h 20 mm inner dimension wi 16 mm hi 16 mm Joint parts weight 21 g Number of segments n 8 Lengths of links l1 l2 l3 l7 l8 125 mm Link angle q1 5 deg q2 175 deg fully com pressed condition External force concentrated load Wy 0 N or 9 8 N In this simulation the self weight of each link is assigned to the corresponding central point Ci represented as a concentrated load We show the simulation result in Fig 13 Having no external load Wy 0 N and only self weight the minimum necessary torque is 9 2 Nm Having 9 8 N of external load the necessary torque increases to 19 Nm The continuous torque capability of motors used is 22 Nm therefore the motor torque should be suffi cient to extend the scissor mechanism Nevertheless the scissor mechanism cannot be extended without exceeding the motors current limits As mentioned in section IV the mechanism cannot be extended upward because it locks Fig 13 The motor torque necessary to extend the 2 DOF scissor mechanism from its fully contracted condition to its fully extended position Calculated for two different external loading conditions no load and 1 kg weight load Next we considered why the scissor mechanism is im possible to extend upward despite the motor torques being theoretically suffi cient Our hypothesis is that the clearance at each joint is the main cause of the singular posture There fore we simulated how the scissor mechanism deforms