A Mechanical Calibration Approach for Binary Parallel Robots

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1 A Mechanical Calibration Approach for Binary Parallel Robots D. Schütz A. Raatz J. Hesselbach Technische Universität Braunschweig Technische Universität Braunschweig Technische Universität Braunschweig Braunschweig, Germany Braunschweig, Germany Braunschweig, Germany Abstract Cost is one of the most important factor for automated production. As there are many handling tasks in production which require only a few destination points, it is reasonable to utilize a low priced robot with a limited workspace. To accomplish such a task, a robot structure being driven by binary actuators and controlled by a programmable logic controller PLC) has been created. Due to the simplicity of this concept the workspace of such robots is discrete. To assure that the endeffector reaches the destination points with the needed accuracy, it is essential to calibrate the robot structure. This contribution deals with a mechanical calibration approach for binary parallel robots. In contrast to usual calibration techniques within the mechanical calibration process, the correction of the parameters must be done by adjusting the relative positions of the structure s components. A parameter correction inside of the robot control is not possible as there are no informations saved about the kinematics of the structure. Keywords: binary robot, mechanical calibration, low-priced robot, digital robot, pick-and-place task I. Introduction Two point pick-and-place applications can be easily done by single binary actuators such as a pneumatic cylinder which is pictured in Fig. 1. A pick-and-place task with a given four point constellation can be implemented either by inflexible mechanisms or expensive robots. This application can also be done sufficiently by a combination of a simple kinematic structure with two binary actuators such as pneumatic cylinders. As these actuators only have two defined states they are easily controllable by a programmable logic controller PLC) which reduces the total costs in comparison to usual robot systems. tions for studying the field of binary robots. Some concepts have the objective to establish binary robots for space activities. For example, Plante et al. has completed research on new lightweight actuators to reduce the total weight of the entire robot [1]. In addition to minimizing the total weight, Sujan and Dubowsky have designed a binary space robot with redundant binary actuators to enhance the robustness of the system [2]. Another application for binary actuation is the elastically averaged parallel manipulator with bistable Dielectric Elastomer Actuators DEA) by Plante and Dubowsky [3]. This manipulator is applied for MRI-guided prostate cancer therapy as it s elastic behavior allows a secured human robot interaction. Binary actuated robots for handling tasks has also been developed by Chirikjian who uses hybrid structures with large numbers of pneumatic cylinders [4]. The goal of this approach is the realization of a binary robot which is able to replace a continuous workspace manipulator. The motivation behind the presented work is the development of a parallel robot with a minimum of binary actuators particularly pneumatic cylinders, which is sufficient for a specific handling task. Furthermore, pneumatic cylinders are highly available and in a wide variety of cylinder diameters and piston strokes. The simple active principle of these binary actuators allows great repeatability in their two defined states and a suitable maximum velocity. Table I shows the usual characteristics of pneumatic cylinders. Description Value Reachable positions 2 Repeatability 0.05 mm Max. velocity 1.5m/s Available cylinder diameter mm Available strokes e.g. 32 mm) mm TABLE I. Characteristics of a low priced pneumatic cylinder. Fig. 1. Low priced binary actuator such as pneumatic cylinders. There are several research groups with different motiva- d.schuetz@tu-bs.de a.raatz@tu-bs.de j.hesselbach@tu-bs.de The combination of two pneumatic cylinders within a parallel kinematic structure has several advantages in relation to a serial configuration. The moved mass of the parallel robot can be comparatively less then that of serial robots. This enables higher velocity and acceleration of the robot. Moreover, the stiffness of the robot is increased, which leads to a better repeatability of the entire robot system. The drawback of using binary actuators is that the robot has a discrete workspace. A robot structure 1

2 with n binary actuators is able to reach m = 2 n destination points. This means that ensuring a good repeatability of this robot is not a guarantee for a sufficient precise pickand-place routine. As the programming of the robot is not possible with the usual teach-in method, it is essential that the robot also comes with a adequate accuracy. In the case that a sufficient accuracy of robots is needed a parameter calibration is applied. Thus, a calibration technique for binary parallel robots is absolutely necessary. Due to the totally different concept in comparison with common robotic systems the calibration method must be customized for binary parallel robots. This contribution deals with a mechanical calibration approach which uses additional components to adjust the kinematic structure of the robot. II. Differences in Parameter Calibration between Continuous and Binary Robots Typically, a kinematic parameter calibration is classified in four steps: Mathematical Model Formulation Measurement Parameter Identification Implementation First, the kinematic structure must be described by a mathematical model. This model must include parameters which can consider production and assembly errors. These errors result in deviations of structure poses of the real robot to its mathematical model. The deviations can be measured, for example, at the tool center point TCP) of the robot. The difference between the actual TCP position P i,a and the nominal TCP position P i,n k)inthei-th position depending on the parameter vector k can be written in a residual r i as shown in Eq. 1. r i k) = P i,a P i,n k) 1) The residual of each measured position can be combined in one vector r. By multiplying this vector with its transposed r T results in a scalar F Eq. 2). F = r T r 2) The scalar F is a typical cost function which can be minimized within an optimization procedure by modifying k. As the mathematical model approaches the actual structure, the cost function F tends to zero. The resulting optimized kinematic parameters k opt can then be implemented into the robot control [5], [6]. Due to the fact that no kinematic parameter values are implemented inside of the PLC of a binary robot, a conventional parameter correction is not possible. It is important to mention that an uncalibrated binary robot may not be able to meet the accuracy requirements of the task. Thus, this paper presents a mechanical calibration approach which allows the direct adjustment of the physical binary robot structure. Therefore, new components are integrated in the kinematic chain of the robot. These additional joints allow the entire structure to be precisely aligned. The mechanical calibration requires a customization of the usual calibration classification in the following manner: Mathematical Model Formulation Measurement Variable Parameter Identification Mechanical Correction III. A Prototype of a Binary Parallel Robot The method of increasing the accuracy by a mechanical calibration should be validated with a prototype of a binary robot. The four point pick-and-place task, which is brought up above, can be performed by two pneumatic cylinders within a RPRPR structure. Where R) stands for a revolute joint and the underlined P) represents an active prismatic joint, which in this case is a pneumatic cylinder. The basic design of this binary parallel robot and its four possible destination points are presented in Fig. 2. P j s 1 s 2 c 1 c 2 A 1 A 2 Fig. 2. The RPRPR structure with binary actuators and its four possible destination points. As the binary parallel robot has a discrete workspace the installation procedure is essential to reach the required positions of a specific task. The given coordinates of the regarded application are the input parameters for an four point synthesis approach, which has already been published [7]. This approach allows the values of the geometrical parameters of the structure to be defined, as listed in Tab. II. Description Parameter Coordinates of the A i = ) x A,i ; y A,i fixed revolute joints Cylinder length c i Stroke of the piston s i Coordinates of the TCP P j = ) x j ; y j TABLE II. Geometrical parameters of the RPRPR structure. The computed parameter values for a given four point constellation P j with j = 1,...,4 can be used to order the pneumatic cylinder with the needed strokes s i. These cylinders can then be implemented in the robot structure with the calculated coordinates of both fixed revolute joints A i and 2

3 the cylinder length c i. Ideally, this installed robot is able to satisfy the requirements of the specific task. Figure 3 shows the closed kinematic chain with the minimal necessary components of the binary parallel robot. Fig. 3. Kinematic chain of the RPRPR structure. Fig. 4. Kinematic chain of the pprpprpprpp structure. A. Extended Kinematic Structure As errors occur during the production of the robot s structural components, as well as at the assembly of the robot structure, the accuracy of the robot may not be sufficient. If the robot is built with the minimal number of links and joints, an adjustment of the structure s parameters is not possible. The discrete nature of the robot s workspace does not allow it to overcome these shortcomings. Thus, some form of reconfiguration needs to be taken into account, in order to improve the accuracy. This demand implies new additional components which allow the adjustment of structure parameters. In addition to there adjustment, a modular design which allow the original structure to be reused in future configurations. This means for the RPRPR structure that regarded geometrical parameters of Tab. II must be variable for installation or for calibration of the binary parallel robot. The robot components for reconfiguration regarded in this contribution can adjust the coordinates of the fixed revolute joints A i with two additional prismatic joints and vary the cylinder length c i with a third additional prismatic joint. The new extended robot structure is here denoted as pprpprpprpp structure where lower case letters stand for adjustable components. Figure 4 illustrates the corresponding closed kinematic chain of Fig. 3 with these new additional robot components for reconfiguration. B. Development and Design of the Binary Parallel Robot In addition to the opportunity to adjust selected geometrical parameters of the robot structure, each part of the structure is developed with a modular design. This allows a reconfiguration into a completely new robot structure with other four point constellations, a larger number of destination points, as well as with different binary actuators in terms of cylinder diameters and piston strokes. The developed component for adjusting A i is a cross table cf. Fig. 5) with determined interfaces for an easy implementation into the presented robot structure. This cross table is fitted between the base and the fixed revolute joints of the regarded robot structure. Fig. 5. Cross table to adjust the fixed joint positions A i. In operation mode the moving parts of the cross table are clamped by screws so that the cross table is completely fixed. To adjust these parameters, the clamping screws must be loosen. Accordingly, the tables can separately be moved by a spindle, which can be rotated manually. The travels of the cross table are measured precisely by removable indicating calipers. The correction of the cylinder length c i is realized by a slide located between the fixed revolute joints and the pneumatic cylinder, which is shown in Fig. 6. The slide can be equipped with cylinders of different 3

4 IV. The Accomplishment of the Mechanical Calibration for Accuracy Enhancement This section deals with the new approach of mechanical calibration of binary parallel robots. The objective is to enhance the accuracy of the robotic system presented in Sec. III. This should be done in the described order of the mechanical calibration Sec. II). Fig. 6. Slide to adjust the cylinder length c i with pneumatic cylinders of different length and type. lengths and types. The relative position of the cylinder to the fixed revolute joint represents the cylinder length c i. The adjustment of this length c i is realized with a bracket on the slide and a simple screw with a fine pitch thread. This screw is connected to the interface plates of the fixed revolute joint and allows a relative adjustment. The travel can analogical be measured with a removable indicating caliper. The plates are able to clamp the slide to fix the adjusted length. The prototype of the binary parallel robot has been built on a breadboard with specific parameter values, which are generated within the four point synthesis algorithm. The revolute joint which is located between the pneumatic cylinders features a additional small pneumatic cylinder and a vacuum gripper to lift objects cf. Fig. 7). This prototype of a binary parallel robot can be applied for a pickand-place task with four destination points which have been discussed in Sec. I. The robot is controlled by a simple PLC. Fig. 7. The Prototype of a Binary Parallel Robot. This test bench has been utilized to validate the approach of the mechanical calibration. A. Mathematical Model Formulation The third step of the mechanical calibration, the variable parameter identification, requires a mathematical model of the robot structure which takes the relevant parameters into account. Due to the simplicity of the kinematic structural model, this step is skipped in favor to other novel content. The relevant parameters of the structure are diveded into the fixed parameters such as the strokes of the pneumatic cylinders s i as well as the parameters which are represented by the additional robot components for reconfiguration, in this case A i and c i. The used kinematic model is equal to the one that is applied inside the four point synthesis algorithm. Due to the mechanical calibration approach it is important that in the function of the Direct Kinematic Problem DKP) of the structure the constant parameters k c and variable parameters k v are distinguished as the constant parameters cannot be changed within the robot control. B. Measurement The next step of mechanical calibration is to provide informations about the real structure position. This has been achieved by a measurement of the endeffector in each of the four actual positions P j,a. This data was gathered using a laser tracker. This measurement system does not fit the entire concept of a low priced pick-and-place robotic system. Other more practical methods to gain data will be the subject of future research. For a validation of the mechanical calibration method it is insignificant that the laser tracker is very high in price. In contrast to the measurement data, the endeffector positions are calculated with the DKP P j,n depending on the nominal parameters k which have been computed within the four point synthesis algorithm. The difference between these coordinates can be written in a residual r j k v ) analog to Eq. 1 which only depends on the variable parameters of the structure, as can be seen in Eq. 3 r j k v ) = P j,a P j,n k v ) 3) and a cost function F can be set up as presented in Eq. 2. This factor F can be applied inside of an optimization algorithm to minimize the differences between the actual and nominal endeffector positions. C. Variable Parameter Identification Due to the non-linear relationship between the endeffector positions P j and the variable parameter of the struc- 4

5 ture k v an iterative optimization algorithm based on least squares is achieved. The well documented Levenberg- Marquardt method is selected as one of its characteristics is a high robustness and it has been proved itself for different calibration techniques [8], [9]. The variable parameter deviations Δk v must be identified which lead to a minimization of the cost function F inside of one iterative loop of the optimization algorithm. Therefore, an identical Jacobian J id is used within Eq. 4 to calculate the variable parameter deviations Δk v subject to the residual vector r, as follows Δk v = J id T J id μi ) 1 Jid T r. 4) Here, μ provides as attenuation coefficient multiplied with the identity matrix I to make sure that the complete matrix of Eq. 4 cannot lose rank [8]. The necessary identical Jacobian J id can be set up in accordance to the numerical finite difference method which approximates the solutions to differential equations by replacing derivative expressions presented by Scales [8]. The computed variable parameter deviation vector Δk v is subtracted from the current variable parameter vector k v which lead to a modified vector k v,mod = k v Δk v. 5) The new parameter set of Eq. 5 is checked whether the resulting residuals can scale the current cost function F down. Thus, the cost function gets smaller during each iterative loop of the optimization algorithm until the value of F falls within a predefined range or the maximum number of loop iterations is reached. In both cases, the optimization algorithm outputs the last qualified variable parameter vector k v,opt which represents the optimal parameter set. The values of this vector can be used to correct the differences of the actual and nominal endeffector positions. D. Mechanical Correction The optimized parameters allow theoretical to correct the deviations between the positions of the mathematical model to the real structure. This means that the integration of these parameters in the DKP would correct the four positions of the model P j,n kv,opt ) P j,a. 6) The destination points P j,n of Eq. 3 are the target points which represent the necessary condition of the specific given task. These points must not be changed during the calibration. As this approach is intended for a mechanical correction of the robot structure by means of manually adjusting the additional robot components, the found optimized parameter set k v,opt must be converted into a displacement of the mechanic adjustable parameters δk v,opt, such that δk v,opt = k v k v,opt. 7) Thus, a correction is possible which allows to minimize the derivations of the given four points to the actual points of the kinematic structure that also enhance the accuracy of the entire binary parallel robot. V. Results This section presents the results of repeatability and accuracy potential of the regarded prototype of the binary parallel robot. The chosen measurement procedures are based on the European Standard EN ISO 9283 Manipulating industrial robots Performance criteria and related test methods which should ensure the comparability of the found characteristic data [10]. Before finding the accuracy of this robot, the proposed mechanical calibration method has been applied. A measurement before the mechanical calibration can derive the repeatability RP of the robot, which should remain constant, independent of the calibration process. In addition, the identification of the robot s accuracy AP is not yielding to a comparable value as it depends on the quality of the robot installation. A. Repeatability The repeatability of a robot can be obtained in accordance with EN ISO 9283 by the measurement of minimum 30 times at point P j. The data of the j-th point at the o-th time is converted to a scalar by using the average values of each coordinate as follows l j,o = xj,o x j y j,o y j z j,o z j. 8) The average value of l j,o l j = 1 p p l j,o 9) o=1 represents together with the triply standard deviation S = p o=1 l j,o l j p 1 the repeatability of the robot in the j-th position. 10) RP j = l j 3S 11) The criteria value of RP j is displayed as radius of the cicles in Fig. 8 where the center of each cicle is the balance point P j,a of the measurement data of the regarded position P j. The largest value of RP j defines the repeatability of the entire robot, which is RP mm. 5

6 Destination Point P1 Destination Point P RP=0.031 mm AP=0.216 mm RP=0.027 mm AP=0.328 mm Destination Point P3 Destination Point P RP=0.036 mm AP=0.241 mm RP=0.020 mm AP=0.172 mm Fig. 8. Experiemental repeatability and accuracy results of the prototype of the binary parallel robot. B. Accuracy The accuracy AP of a robot is defined by the derivation between the nominal position P j,n and the average value of the actual positions P j,a. Let x j,n, y j,n and z j,n be the coordinates of the nominal position P j,n then the accuracy AP j of the j-th point can be derived from AP j = xj x j,n yj y j,n zj z j,n. 12) Figure 8 illustrates the derivation of the nominal point P j,n to the balance point P j,a of the measurement data and the corresponding values of AP j. Analogous to the repeatability, the accuracy AP of the entire binary parallel robot after the mechanical calibration is given by the largest value of AP j. Thus, the robot allows a accuracy of AP mm. VI. Conclusions and Outlook This contribution presents a basic approach for the calibration of binary parallel robots. This form of robotic system allows to use binary actuators such as conventional pneumatic cylinders within the kinematic structure of the robot. The control of binary actuators is rudimental and can be achieved using a standard PLC. Due to the PLC has no information saved about any geometric parameter of the robot structure, a typical calibration cannot be accomplished to enhance the accuracy of a binary parallel robot. Thus, a new approach of mechanical calibration is introduced in this publication, which requires additional robot components inside of the kinematic structure. These components must enable the adjustment of geometrical parameters inside of the kinematic chain of the robot. Hence, only these parameters are variable during the optimization algorithm of the parameter identification. After illustrating the basic steps of mechanical calibration, the novel approach has been validated at a prototype of a binary parallel robot with an RPRPR structure. The presented results of the repeatability and accuracy of the calibrated prototype of a binary parallel robot show that a low priced robotic system with binary actuators and a PLC can be suitable for simple pick-and-place tasks. The search of a low priced measurement system for the mechanical calibration of these robotic system will be one of the next steps in the binary parallel robot investigation, as the laser tracker does not match the price class of this concept. Additional kinematic structures will be analyzed 6

7 to make a statement into how many binary actuators make sense inside of a robot structure for handling tasks. References [1] Plante J.S., Santer M., Dubowsky S., and Pellegrino S. Compliant bistable dielectric elastomer actuators for binary mechatronic systems. In Proceedings of the ASME Conference on Mechanisms and Robotics [2] Sujan V. and Dubowsky S. Design of a lightweight hyper-redundant deployable binary manipulator. ASME Journal of Mechanical Design, 1261):29 39, [3] Plante J.S. and Dubowsky S. The calibration of a parallel manipulator with binary actuation. In Advances in Robot Kinematics: Analysis and Design, pp Springer, Dordrecht, [4] Chirukjian G. Kinematic synthesis of mechanisms and robotic manipulators with binary actuators. Journal of Mechanical Design, 117: , [5] Zhuang H. and Roth Z. Method for kinematic calibration of stewart platforms. Journal of Robotic Systems, 103): , [6] Bernhardt R. and Albright S.L. Robot calibration. Chapman & Hall, London, 1st edition, [7] Schütz D., Raatz A., and Hesselbach J. The development of a reconfigurable parallel robot with binary actuators. In J. Lenarčič and M.M. Stanišić, eds., Advances in Robot Kinematics, pp Springer ScienceBusiness Media B.V, Dordrecht, [8] Scales L.E. Introduction to non-linear optimization. Macmillan computer science series. Macmillan, London, [9] Last P., Budde C., Schütz D., Hesselbach J., and Raatz A. Parallel Robot Calibration by Working Mode Change. In Advances in Robot Kinematics: Analysis and Design, pp Springer, Dordrecht, [10] European Committee for Standardization CEN). Manipulating industrial robots - Performance criteria and related test methods EN ISO 9283),

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