Article
Unified Parameterization and Calibration of Serial, Parallel, and
Hybrid Manipulators
Benjamin L. Moser
1,
* , Joshua A. Gordon
2
and Andrew J. Petruska
1
Citation: Moser, B.L.; Gordon, J.A.;
Petruska, A.J. Unified
Parameterization and Calibration of
Serial, Parallel, and Hybrid
Manipulators. Robotics 2021, 10, 124.
https://doi.org/10.3390/
robotics10040124
Academic Editor: Giovanni Boschetti
and João Miguel da Costa Sousa
Received: 30 September 2021
Accepted: 12 November 2021
Published: 17 November 2021
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1
M3 Robotics Laboratory, Department of Mechanical Engineering, Colorado School of Mines,
Golden, CO 80401, USA; apetruska@mines.edu
2
National Institute of Standards and Technology, Boulder, CO 80305, USA; josh.gordon@nist.gov
* Correspondence: bmoser@mines.edu
Abstract:
In this work, we present methods allowing parallel, hybrid, and serial manipulators to
be analyzed, calibrated, and controlled with the same analytical tools. We introduce a general
approach to describe any robotic manipulator using established serial-link representations. We use
this framework to generate analytical kinematic and calibration Jacobians for general manipulator
constructions using null space constraints and extend the methods to hybrid manipulator types
with complex geometry. We leverage the analytical Jacobians to develop detailed expressions for
post-calibration pose uncertainties that are applied to describe the relationship between data set size
and post-calibration uncertainty. We demonstrate the calibration of a hybrid manipulator assembled
from high precision calibrated industrial components resulting in 91.1
µ
m RMS position error and
71.2
µ
rad RMS rotation error, representing a 46.7% reduction compared to the baseline calibration of
assembly offsets.
Keywords:
parallel manipulators; robot kinematic calibration; hybrid manipulators; pose uncertainty
1. Introduction
Kinematic calibration is the process in which a parameterized mathematical model
used in planning and control operations is adjusted to minimize pose-error. Parallel manip-
ulators, or Parallel Kinematic Machines (PKM), of varying degrees of freedom (DoF) are
well-suited for high load applications requiring precision motion and have been utilized in
applications such as pick-and-place [
1
], drone manipulators [
2
], and space telescope mirror
adjustment [
3
]. Parallel manipulators are advantageous with respect to physical properties
of manipulator stiffness and deflection, but have a significantly reduced workspace volume
relative to serial manipulators. Hybrid manipulators incorporating elements from both
parallel and serial manipulators combine these positive attributes and have been pro-
posed for novel mechanism design [
4
], near-field antenna measurements [
5
] and medical
robots [6]
. Although the majority of the parallel mechanisms in the literature incorporate
coincident passive joints in the form of universal or spherical joints at the top and bottom
of a prismatic leg similar to the popular 6 DoF Stewart–Gough configuration [
7
,
8
], recent
work has modified the design to use offset universal joints to increase stiffness and reduce
manufacturing constraints [
9
]. This 6-RRRPRR parallel manipulator configuration with
offset R-R joints, with individual parallel members constructed of rotary (R) and prismatic
(P) joints listed in order, requires more general kinematic and calibration approaches than
typically used with Stewart Platform designs due to the removal of the geometric sim-
plifications permitted by the coincident passive joints and is of particular interest within
this work. We present constraint-based approaches to solve the kinematics and calibration
problems for strictly parallel manipulators and hybrid parallel-serial manipulators, such as
the example pictured in Figure 1, with the general construction shown in Figure 2.
Robotics 2021, 10, 124. https://doi.org/10.3390/robotics10040124 https://www.mdpi.com/journal/robotics
