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Edited by: Nikola Miskovic, University of Zagreb, Croatia

Reviewed by: Ahmed Chemori, CNRS, France; Arnaud Leleve, Université de Lyon, France; Ivana Palunko, University of Dubrovnik, Croatia

Specialty section: This article was submitted to Robotic Control Systems, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A control law combining motion performance quality and low stiffness reaction to unintended contacts is proposed in this work. It achieves prescribed performance evolution of the position error under disturbances up to a level related to model uncertainties and responds compliantly and with low stiffness to significant disturbances arising from impact forces. The controller employs a velocity reference signal in a model-based control law utilizing a non-linear time-dependent term, which embeds prescribed performance specifications and vanishes in case of significant disturbances. Simulation results with a three degrees of freedom (DOF) robot illustrate the motion performance and self-regulation of the output stiffness achieved by this controller under an external force, and highlights its advantages with respect to constant and switched impedance schemes. Experiments with a KUKA LWR4+ demonstrate its performance under impact with a human while following a desired trajectory.

A key challenge for the successful introduction of robots in human centered environments, as domestic assistants or co-workers, is the concurrent resolution of the issues of task performance and safety for the coexisting human (De Santis et al.,

Humans cope superbly with collisions and contact uncertainty by flexibly modulating their arm/hand compliance. Compliance protects the human from excessive forces during impact and can be achieved in robots either passively by using flexible components in the robot’s structure or actively by the controller. Passive compliance is very important for the reduction of the initial collision force, which is responsible for the so-called pre-collision safety (Heinzmann and Zelinsky,

As service robots have to perform useful tasks for humans in a dynamic and uncertain environment, quality of performance is desired. The prescribed performance control methodology introduced in Bechlioulis and Rovithakis (

The aim of this work is to concurrently address the competing requirements of motion performance and compliance under impact by designing a control scheme that achieves prescribed performance in nominal operation (high stiffness), a compliant reaction at impact (low stiffness) and smooth transition between the two modes. The system self-regulates the output stiffness according to the disturbance level without explicit collision detection and control switching which are subjective to delays and jeopardize performance and stability. The feedback controller is model-based assuming knowledge of the robot’s model and measurements of joint positions and velocities. The paper is organized as follows: In Section

Consider a first-order integrator scalar system of a tracking error

For system [equation (_{t}_{→∞}_{∞} > 0 called performance function. A candidate performance function is the exponential
_{0}, _{∞}, _{0} = _{∞} represents the maximum allowable size of the output error

By considering the modulated error ^{c}

In this work, we combine both requirements of nominal performance and enhanced safety operation under impact by proposing a new controller design based on a transformation, which defines the following smooth, non-decreasing, non-linear, surjective mapping of the modulated error domain:

Hence, the transformation is strictly increasing in ^{c}

Notice that in place of equation (

The artificial potential induced by such saturated transformation:

Let us further define:

Notice that

Property [equation (

Function

Using _{s}_{t}_{→∞}α(

R

Substituting control input [equation (

Differentiating

For the unforced non-autonomous system [equation (^{c}_{s}ξ_{s}

In the presence of a bounded input

T

Then, there exists an invariant set _{0} ⊂ _{0} guarantees a nominal performance error evolution in the sense of equation (

_{1}(|_{2}(|_{1}, _{2} are class 𝒦 functions, and

If

Next, we simplify the analysis by considering odd _{1}, _{2} satisfy 0 < _{1} < _{2} < 1 as shown in Figure

Hence, defining
_{0} then _{0} ⊂ ^{+}, which implies that _{0} is invariant and the system remains in nominal performance operation.

For the specific case of the candidate transformation function [equation (_{0} illustrated in Figure

R_{0} of the performance function should be selected such that equation (

R_{s}_{s}_{e}

Since our objective is a robot-control design, which complies with large disturbances, there is no need of choosing high values for _{s}_{t}_{→∞}

Consider a _{q}_{e}^{3}, _{e}_{e}

The robot dynamic model can be written as follows:
_{d}^{3}, _{d}_{e}_{d}

For simplicity and without loss of generality, we proceed by considering the position tracking problem. Thus, our objective is to design a state feedback control law, in order to force the robot’s end-effector position _{e}_{d}_{i}_{i}

T_{d}^{3} and performance functions _{i}_{i}_{ei}_{di}_{i}_{i}_{i}_{si}_{i}_{i}_{v}^{+}(^{+}(^{−1}(_{r}^{3} having _{ri}

Consider now the positive definite radially unbounded function:
_{m}_{M}

Let _{v}_{v}

Defining the region _{q}_{d}_{q}

Given equation (_{q}_{d}_{p}_{q}_{p}

Multiplying with the robot Jacobian _{s}_{i}_{si}, k_{i}_{i}_{i}_{p}_{ext}_{d}^{T}_{ext}_{p}_{d}_{v}

R_{i}_{i}_{i}

R

We consider a three DOF rotational joint spatial robotic manipulator with link masses _{1} = _{2} = _{3} = 1 kg, link lengths _{2} = _{3} = 0.5 m, and link inertias _{x}_{2} = _{x}_{3} = _{z}_{1} = 4.15 × 10^{−4} kg m^{2}, _{y}_{2} = _{z}_{2} = 2.1 × 10^{−2} kg m^{2}, and _{y}_{3} = _{z}_{3} = 0.39 × 10^{−2} kg m^{2}. The robot is initially at rest at the position _{e}^{T}^{T}_{df}^{T}_{do}^{T}^{T}_{i}_{i}_{0} = 0.02 set high enough to ensure initialization within the invariant set _{0} for a range of disturbance magnitudes, _{i}_{∞} = 10^{−3} corresponding to an accuracy of 1 mm and _{i}_{v}_{3}, with _{3} being the identity matrix of dimension 3, _{si}_{i}

We initially consider an impact force _{ext}_{E}_{stiff}) via a series of simulation runs with impact forces of various amplitudes in the range of 5–60 N with a step of 1 N. The stiffness values are calculated by the ratio of the pulse amplitude _{E}_{E}_{E}_{stiff}) corresponding to the nominal performance with high stiffness values and impact reaction modes with low stiffness values (Figure _{i}_{p}^{−3} m as prescribed).

_{x}_{y}_{z}

Next, we consider joint disturbances arising by an uncertain gravity vector model. Joint disturbances arising from a partially compensated gravity 0.6

Last, we have simulated the case of an impact with an environment modeled as a spring with stiffness of 1000 N/m, obstructing the motion of the arm for 0.5 s. For comparison purposes, we have simulated the case of the robot being under the impedance control scheme of high targeted stiffness as well as a switched impedance between the high and a low stiffness with a delay of 0.001 s (an ideal case examined for comparison purposes) and 0.2 s from the moment of impact in order to account for the time needed for the impact detection and reaction response (a practical switched impedance case). Stiffness values were selected from those appearing in the two modes of operation for the proposed controller and were 28,000 and 600 N/m, respectively. Figure _{∞} results in a lower force peak appearing earlier; with a very small value (_{∞} = 10^{−9}) the interaction force behaves like the ideal switch case.

_{∞}.

Experiments are conducted with a KUKA LWR4+ 7 DOF robotic manipulator. The control law of equation (_{i}_{0} = 0.01, _{i}_{∞} = 0.005, and _{i} = 20, _{si}_{i}_{v}_{d}^{T}^{T}

Next, we consider the case of a human standing in the robot’s way causing an unintentional contact on his back (Figure ^{T}^{T}_{df}^{T}

It is well-known that setting a low-desired stiffness in a conventional impedance controller for safety reasons adversely affects performance. On the other hand, high-targeted stiffness in impedance control can achieve a certain tracking quality and robustness but adversely affects human and robot safety. By contrast, the proposed controller addresses both objectives of motion performance and enhanced safety in one scheme. With regards to motion performance quality, it achieves prescribed performance tracking both in transient and steady state respecting performance bounds under disturbances up to a tunable level conceptually separating the nominal operation and the impact reaction modes. Performance in the nominal operation mode has been demonstrated under disturbances arising by partially compensating gravity in simulations and by ignoring the feedforward control terms related to inertial and Coriolis forces in experiments. Thus, the model-based structure in the inner loop does not jeopardize performance. The threshold of the disturbances allowing the operation in the nominal performance mode can be regulated by changing the value of _{i}

Enhanced human and robot safety is on the other hand achieved by operating in the impact reaction mode where the apparent output stiffness is characterized by low values as compared to the high stiffness values characterizing the nominal operation. Stiffness drop was demonstrated in both simulations and experiments where force values shown in Subsections _{∞}. Notice, however, that _{∞} values cannot be chosen arbitrarily small since they should reflect the accuracy achieved by the measurement device, and ensure that the steady state performance zone is wide enough to accommodate the measurement noise.

This work proposes a controller in which the robot output stiffness is self-regulated according to the disturbance magnitude. Moreover, the error is shown to evolve within a predefined performance region in nominal operation mode exhibiting robustness to disturbances up to a tunable threshold. The system reacts stably by reducing its output stiffness under bigger disturbances like those arising from impact, returning to the nominal operation after the disturbance vanishes. The controller achieves a continuous and smooth transition between the two modes without switching, eliminating the need for separate collision detection, and reaction strategies. Simulations and experimental results demonstrate the enhanced safety achieved by the proposed controller under impact with initial impact force magnitudes connected to the prescribed steady state error bounds of the nominal operation mode. Comparison with a practical switched impedance control scheme shows that the proposed control law achieves a slightly better performance without making use of any switching. Future work will further investigate safety under unintentional contacts by taking into account disturbance frequency content.

I have developed the basic idea for the proposed controller together with my co-author YK, and I have supervised and coordinated the simulated and experimental work, which was performed by my PhD students and co-authors LD and DP.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research is co-financed by the EU-ESF and Greek national funds through the operational program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program ARISTEIA I under Grant PIROS/506. This work is also partially funded by the Swedish Research Council (VR).