On the Repeatability of Planar 2R Manipulators with Rotary Encoders Mathias Brandst¨ otter, Christoph Gruber, and Michael Hofbaur Institute of Automation and Control Engineering UMIT A-6060 Hall in Tirol, Austria Email: {mathias.brandstoetter & christoph.gruber & michael.hofbaur}@umit.at Abstract—The performance of position control systems for manipulators directly depends on the accuracy of its measure- ment system. Robots with revolute joints typically use rotary encoders for measuring the joint angles. The quantization in the joint angle measurements leads to a discretization of the configuration space and thereby limits accuracy and repeata- bility. Configuration space coordinates transform to workspace coordinates via nonlinear direct kinematic maps. The equidistant discretization of the configuration space therefore results in a non- equidistant discretization of the workspace. One effect of this non-equidistant workspace discretization is that repeatability is not constant over the workspace but depends on the configuration of the robot. In this work, we present a measure for repeatability that is able to deal with this configuration-dependence. Based on this measure, we are able to provide regions of maximal repeatability, which are regions where the positioning error due to discretization is minimal. The methodology is demonstrated for planar 2R manipulators. I. INTRODUCTION For a serial robot with revolute joints and angular encoders, the computation of the location of the end-effector is done using the direct (forward) kinematics, which is a map from joint angles to end-effector positions. These direct kinematics depend on the geometry of the robot and its parameters, like for example the link lengths. For several practical reasons, geometry is never perfectly accurate, resulting in a deviation between the real position of the robot and the position com- puted via direct kinematics. Such a deviation is called accuracy in literature [1]. Repeatability describes how close the end-effector of a robot manipulator can come to a point where it has already been before. Deviations in the geometry therefore have no influence on repeatability. Deficiencies of repeatability mainly originate from joint sensor errors, non-rigidity of the joints, thermal effects, etc. [2]. In this work, we consider the repeata- bility of perfectly rigid serial planar 2R manipulators. We treat the deficiencies due to the finite resolution of rotary encoders, known as control resolution [3]. Although we restrict ourselves to such simple robots and allow only one source for error, we will show that the following seemingly intuitive statement is not true in general: “Repeatability is maximal when the end- effector is closest to the base”. But before getting into detail with that, let us introduce some more terms and related work. A lot of research deals with positioning errors defined as distance between a desired and a real end-effector position. The impact of geometric distortions on such positioning errors are, amongst others, examined in Balli and Chand [4], Shiakolas et. al. [5], and Veitschegger and Wu [6]. Also probabilistic approaches ([2], [7], [8]) using the Jacobian matrix to map the joint errors to end-effector location errors were examined. Another common term used in literature is spatial resolu- tion, which is understood as the distance between two neigh- boring discrete end-effector positions in the workspace. For manipulators with revolute joints, this spatial resolution is not constant over the workspace. Since repeatability of real robots is lower-bounded by the spatial resolution, also repeatability depends on the actual configuration. Breth´ e [9], for example, states that spatial resolution is high near the singularities of a robot. In this paper, we will show that, repeatability may still be low in these areas. Another measure for accuracy was introduced by Veryha and Kurek [10]. They take the Euclidean distance in the workspace between two specific neighboring discrete configurations as positioning accuracy measure. We will show that their selection of these two specific neighbors can lead to invalid conclusions. Briot and Bonev [11] compare 2-DOF planar serial and parallel robots. For comparison, they use the maximum workspace distance between two discrete configurations as accuracy measure. Our main contribution is a measure for repeatability that bases on a similar approach. Moreover, this measure is con- sistent with the methods from Merlet [12] for parallel robots. For simple robots like the planar 2R manipulator, the measure can be evaluated analytically, demonstrated in this paper. For robots with more complex structure, numerical computation is possible. Furthermore, our method allows to find regions of maximum repeatability in the workspace. II. KINEMATICS The forward kinematic map φ of a two-link planar manip- ulator in a reference frame (O, x, y) is given by: φ : R 2 → R 2 q → φ(q) :=  φ 1 (q) φ 2 (q)  =  x p y p  = =  r 1 cos q 1 + r 2 cos(q 1 + q 2 ) r 1 sin q 1 + r 2 sin(q 1 + q 2 )  , (1) 978-1-4799-2722-7/13/$31.00 c 2013 IEEE