Journal of Robotics

Volume 2018 (2018), Article ID 2412608, 9 pages

https://doi.org/10.1155/2018/2412608

## On the Direct Kinematics Problem of Parallel Mechanisms

Workgroup on System Technologies and Engineering Design Methodology, Hamburg University of Technology, 21073 Hamburg, Germany

Correspondence should be addressed to Arthur Seibel

Received 26 October 2017; Revised 1 January 2018; Accepted 9 January 2018; Published 12 March 2018

Academic Editor: Gordon R. Pennock

Copyright © 2018 Arthur Seibel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The direct kinematics problem of parallel mechanisms, that is, determining the pose of the manipulator platform from the linear actuators’ lengths, is, in general, uniquely not solvable. For this reason, instead of measuring the lengths of the linear actuators, we propose measuring their orientations and, in most cases, also the orientation of the manipulator platform. This allows the design of a low-cost sensor system for parallel mechanisms that completely renounces length measurements and provides a unique solution of their direct kinematics.

#### 1. Introduction

A typical six-degrees-of-freedom parallel mechanism consists of a (fixed) base platform and a (movable) manipulator platform. The position and orientation (also known as pose) of the manipulator platform are commanded by fixing the distances between points on the base platform and points on the manipulator platform, where . There may be different ways for realizing such a mechanism. The most common one is to use six linear actuators for connecting the platforms together.

Determining the pose of the manipulator platform from the linear actuators’ lengths (also known as direct kinematics problem) generally leads to a system of algebraic equations that has at most 40 different solutions [1–8]. This number of solutions can be further reduced by introducing additional constraints, for example, combinatorial or planarity constraints [9]. Nonetheless, a closed-form solution cannot be realized by only measuring the lengths of the linear actuators.

Current sensor concepts for solving the direct kinematics problem can be basically classified into two groups [10]. The first group consists of using the minimal number of sensors, in our case, six length sensors, and then including additional numerical procedures to uniquely identify the parallel mechanism’s pose [11–20]. These procedures, however, are generally not real-time capable, require an initial estimate of the solution, and may exhibit convergence problems or even converge to a wrong solution. The requirement of an initial solution estimate is especially then problematic when starting the mechanism at an arbitrary pose.

In contrast, the second approach consists of adding extra sensors for obtaining additional information about the parallel mechanism’s state [21–28]. These can be, for example, angular sensors that are placed on the base or the manipulator platform joints or linear and/or angular sensors that are placed on supplementary passive legs. Here, the number and location of the sensors must be carefully chosen because, otherwise, this may cause specific problems such as workspace limitations due to the passive leg or joint arrangement. Furthermore, using different sensor types leads to a higher complexity and may even negatively affect the performance due to possible time delays and/or conflicting measurement values.

For this reason, in order to provide a unique solution of the direct kinematics problem without using additional numerical procedures or sensors, instead of measuring the lengths of the linear actuators, we propose measuring their orientations and, if necessary, also the orientation of the manipulator platform. The orientations of the linear actuators and the roll-pitch orientation of the manipulator platform can be measured, for example, by acceleration sensors with three axes, and the measurement of the yaw orientation of the manipulator platform can be realized, for example, by using a magnetic sensor [29].

The remainder of this paper is organized as follows. In Section 2, a classification of six-degrees-of-freedom parallel mechanisms based on the number of base and manipulator platform joints as well as combinatorial classes is introduced. In Section 3, we investigate if a closed-form solution for the direct kinematics problem of the mechanism types presented in Section 2 is possible by only measuring the orientations of the linear actuators. In Section 4, for the mechanism types where a closed-form solution of the direct kinematics problem is not possible by only measuring the linear actuators’ orientations, we also include the information about the roll-pitch orientation of the manipulator platform. In Section 5, we discuss the last remaining case where also the information about the manipulator platform’s yaw orientation is included. In order to complete our systematic investigation, in Section 6, we extend our results to three-degrees-of-freedom planar mechanisms. Section 7 discusses some practical considerations regarding the sensor selection and implementation of the proposed algorithms in a real-time control. Finally, in Section 8, our results are summarized and discussed.

Throughout the paper, we use the following notation, referring to Figure 1. The body-fixed frame of the base platform is denoted as and the body-fixed frame of the manipulator platform as . The position vector of the th joint of platform is denoted as and the connection vector between the joints and of platforms and as with . Using inverse kinematics, this vector can be determined fromwith respect to platform . Here, denotes the rotation matrix from frame into frame , and is the vector connecting the origins of platforms and . The roll, pitch, and yaw angles of the manipulator platform shall be denoted as , , and , and the direction, or orientation, of is referred to as , which has unit length.