Guaranteed manipulator precision via interval analysis of inverse kinematics

The paper presents a new methodology for solving the inverse problem of manipulator precision design. Such design problems are often encountered when the end-effector uncertainty bounds are given, but it is not clear how to allocate precision bounds on individual robot axes. The approach presented in this paper uses interval analysis as a tool for uncertainty modelling and computational analysis. In prior work, the exponential formulation of the forward kinematics map was extended to intervals. Here, we use this result as an inclusion function in the computation of solutions to set-valued inverse kinematic problems. Simulation results are presented in two case studies to illustrate howwe can go from an uncertainty interval at the end-effector to a design domain of allowable uncertainties at individual joints and links. The proposed method can be used to determine the level of precision needed in the design of a manipulator such that a predefined end-effector precision can be guaranteed. Also, the approach is general as such it can be easily extended to any degree-of-freedom and kinematic configuration. ∗Address all correspondence to this author. INTRODUCTION The success of automated assembly by robotic manipulators is highly dependent on the precision of the positioning mechanisms employed in the kinematic design of the robot. The importance of precision becomes more prominent when the desired operational accuracy is in micro/nano scale as in micro-assembly and nano-manipulation applications. Most of the parametric uncertainties that are negligible in conventional robotics become the predominant error sources in micro/nano applications. With the emergence of micro/nano-robotics in the last decade, there is a growing demand for design guidelines describing how to build these robots based on application specific precision criteria. Since the current methodology have not addressed this problem well, we propose a new approach to the kinematic analysis and design of robots using interval analysis. In late 80’s and early 90’s, modelling of errors in robot kinematics was a popular topic [1–6]. However, these works are limited to analysing the effects of general error transformations and do not address how they come about or how to choose or design the parameters of a manipulator for a given end-effector preci1 Copyright © 2013 by ASME sion. On the other hand, mechanical design of multi-axis machines based on the accuracy required at the tool tip has been investigated for precision machine design. A concept called error budget was proposed to account for the effect of each error source on the tool accuracy [7, p. 61], [8]. Using first order and small angle approximations to simplify the homogeneous transformation matrices describing geometric errors in the mechanism of a machine, an overall kinematics map can be created to analyse the effects of error terms on the tool accuracy. The results of this analysis provide the designer with insight into how to allocate mechanical tolerances to the individual system components. However, there is no systematic way of doing this allocation due to the fact that this type of an inverse problem is difficult to pose and solve. A recent work in [9] studies modular robotic chains made up of individual axes with link/joint uncertainties for several different configurations. It was shown that some of the kinematic configurations provide more successful operation. Also, joint position uncertainties were shown to manifest themselves as end-effector inaccuracy in different magnitudes depending on the kinematic design. The analysis of error propagation in this work was done using Monte Carlo simulations which suffer from inability to provide guaranteed closed form solutions since Monte Carlo method can only solve for sample points and produce sets of points in the solution space. Finding guaranteed solutions to a set of equations with uncertain data is possible with interval analysis. Interval analysis is a mathematical tool for computation of rigorous bounds on solutions to ideal model equations when the input arguments of the model are represented as intervals instead of point values. It extends the model equations to the interval domain and allows for analytical and computational handling of uncertain data without having to assume a distribution for it or to sample it. It also helps avoid the complex mathematical formulations involving distribution functions [10]. Intervals were used in [11] to model uncertain physical parameters of a robot and to find its forward kinematics map using Denavit-Hartenberg (D-H) notation. Optimization of the D-H parameters was done to minimize volumetric end-effector error while optimizing the cost of precision. While this work nicely approaches the problem from a mechanism design perspective, it fails to consider the fact that a mechanism design cannot be optimized practically without considering the uncertainty in joint positions. Interval analysis was also used in [12] to find the multiple solutions to the forward kinematics problem of parallel robots. Then, [13] extends the method to finding the robot parameters that guarantee a singularity-free workspace. In our prior work [14], the exponential formulation of the forward kinematics map for serial manipulators was extended to intervals. This makes it possible to use interval analysis to find guaranteed precision bounds on the end-effector pose given the uncertainty of the kinematic parameters. The contribution of the current paper is that we now propose a new method that can solve the inverse problem of bounding the allowable uncertainty in kinematic parameters of a manipulator based on given end-effector precision specifications. Besides precision machine designers, this method bears an importance for those roboticists who have to design a manipulator using elementary building blocks. For instance, custom design of multi-axis precision manipulators using individual single-axis stages is a common practice in the micro-assembly area [15–17]. The cost of such stages increase significantly with the increase in motion precision. For a given application, therefore, determining the level of precision required in each axis is an important yet insufficiently addressed consideration. Next section provides an overview of [14], the forward kinematics problem with uncertain parameters. Then, we discuss how the precision manipulator design problem can be posed via interval analysis of inverse kinematics. We present simulation results illustrating the use of our method for a simple 2-link manipulator and a 3-DOF PPR stage, a commonly used robot in precision assembly. Finally, the paper is concluded with references at the end. FORWARD KINEMATICS WITH UNCERTAIN JOINT PA-