Citation: Rojas, R.A.; Giusti, A.;
Vidoni, R. Online Computation of
Time-Optimization-Based, Smooth
and Path-Consistent Stop Trajectories
for Robots. Robotics 2022, 11, 70.
https://doi.org/10.3390/
robotics11040070
Academic Editor: Dario Richiedei
Received: 23 May 2022
Accepted: 21 June 2022
Published: 1 July 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Online Computation of Time-Optimization-Based, Smooth and
Path-Consistent Stop Trajectories for Robots
Rafael A. Rojas
1,2,
* , Andrea Giusti
2
and Renato Vidoni
1
1
Faculty of Science and Technology, Free University of Bozen/Bolzano, Piazza Universita 1,
39100 Bolzano, Italy; renato.vidoni@unibz.it
2
Fraunhofer Italia Research, Via A.-Volta 13A, 39100 Bolzano, Italy; andrea.giusti@fraunhofer.it
* Correspondence: rafael.rojas@unibz.it
Abstract:
Enforcing the cessation of motion is a common action in robotic systems to avoid the
damage that the robot can exert on itself, its environment or, in shared environments, people. This
procedure raises two main concerns, which are addressed in this paper. On the one hand, the stopping
procedure should respect the collision free path computed by the motion planner. On the other hand,
a sudden stop may produce large current peaks and challenge the limits of the motor’s control
capabilities, as well as degrading the mechanical performance of the system, i.e., increased wear.
To address these concerns, we propose a novel method to enforce a mechanically feasible, smooth and
path-consistent stop of the robot based on a time-minimization algorithm. We present a numerical
implementation of the method, as well as a numerical study of its complexity and convergence.
Finally, an experimental comparison with an off-the-shelf stopping scheme is presented, showing the
effectiveness of the proposed method.
Keywords: trajectory planning; optimization; robot stopping scheme; stopping trajectory
1. Introduction
Many robotic applications are subject to unforeseen situations that can endanger the
robot itself, its environment or people sharing its workspace. When this happens, one
possible solution is to eliminate the source of risk, i.e., the robot’s motion. Rather than an
arbitrary stop trajectory, it is desirable that the cessation of motion is satisfied at least three
important characteristics of the original trajectory. First, it must be feasible with respect
to the robot’s actuator limits. Second, it has to satisfy the design criteria of the original
trajectories. Finally, it has to preserve the original path in order to avoid possible collisions.
Among different design criteria for trajectories, the enforcement of smoothness through
the minimization of the jerk has a significant place in the robotics community. This criterion
goes back to [
1
,
2
] and has been reformulated under different settings [
3
–
9
]. Minium jerk
motions reduce the controller’s tracking error [
1
], limit the vibratory content and the conse-
quent mechanical wear on the robotic system [
6
] and successfully model human arm/hand
motions [
2
,
10
,
11
]. In addition, it has been proven that these motions improve the human
subjective acceptance of the robot [
12
], showing a potential application in collaborative
robotics [8,9].
In the literature, path-consistent stop trajectories have been exploited to develop
collaborative robotics applications under the speed and separation monitoring modality, in
agreement with the ISO 10218 and ISO/TS 15066 [
13
–
15
]. In such a modality, the robot’s
motion is tuned through a perception system in order to ensure a protective separation
distance from the human [
16
–
18
]. Non-path-consistent stop trajectories are used in [
19
,
20
].
In order to achieve path consistency, the authors of [
21
–
26
] implemented different
techniques inside the control loop to dynamically choose the linear scaling of the desired
trajectory. Multiple safety criteria are also considered in [
27
] and verified online to formally
Robotics 2022, 11, 70. https://doi.org/10.3390/robotics11040070 https://www.mdpi.com/journal/robotics
