# Improved impedance/admittance switching controller for the interaction with a variable stiffness environment

*Complex Eng Syst*2022;2:12.

## Abstract

Hybrid impedance/admittance control aims to provide an adaptive behavior to the manipulator in order to interact with the surrounding environment. In fact, impedance control is suitable for stiff environments, while admittance control is suitable for soft environments/free motion. Hybrid impedance/admittance control, indeed, allows modulating the control actions to exploit the combination of such behaviors. While some work has addressed the proposed topic, there are still some open issues to be solved. In particular, the proposed contribution aims: (i) to satisfy the continuity of the interaction force in the switching from impedance to admittance control when a feedforward velocity term is present; and (ii) to adapt the switching parameters to improve the performance of the hybrid control framework to better exploit the properties of both impedance and admittance controllers. The proposed approach was compared in simulation with the standard hybrid impedance/admittance control in order to show the improved performance. A Franka EMIKA panda robot was used as a reference robotic platform to provide a realistic simulation.

## Keywords

*,*interaction control

*,*industry 4.0

*,*industrial robotics

## 1. INTRODUCTION

### 1.1. Context

Compliant control ^{[1]} has been widely employed to establish a stable and controlled interaction between a robot and the surrounding environment. While different implementations and approaches can be found in the state-of-the-art to deal with different applications ^{[2-4]}, impedance and admittance controllers ^{[5]} are the most investigated strategies to deal with (partially) unknown environments ^{[6-8]}. While impedance control is suitable to control the interaction between the robot and a stiff environment, admittance control performs better when the robot interacts with a soft environment ^{[9]}. Indeed, a control framework capable of combining both controllers would improve the interaction control performance. It would allow switching between the right behavior for the interaction control considering soft environments (or free motion phases, in which the environment stiffness can be considered null) and for the interaction control considering stiff environments. Even though the modulation of such control behaviors has been addressed by the variable impedance control (*i.e.*, to tune the controller parameters to deal with different environments/task phases ^{[10, 11]}), hybrid control frameworks have also been developed to combine the capabilities of both impedance and admittance controllers. In the following, the state-of-the-art hybrid controllers are analyzed to highlight the related open issues.

### 1.2. Related works

The complementary performance of impedance and admittance controllers is qualitatively shown in Figure 1. A hybrid controller that combines the behaviors of both impedance and admittance control would result in improved interaction control performance and would allow implementing a generalized interaction controller capable of dealing with any interaction environment.

Ott *et al.*^{[12]} summarized the hybrid control framework in ^{[12, 13]}, which enhances the performance of interaction control in the whole spectrum of environment stiffness values. The proposed approach consists of applying the impedance and admittance controllers alternatively by exploiting a switching mechanism defined by a switching period parameter *et al.*^{[14]} and Mei *et al.*^{[15]} applied the hybrid control framework on a simulated *et al.*^{[14]} introduced a neural network (NN) for the choice of *et al.*^{[15]} introduced a damping term *et al.*^{[16]} proposed an adaptive impedance controller using function approximation techniques where the system uncertainties are represented as basis functions and the aim is to converge to target impedance parameters with lower computational requirements. Other approaches have been proposed for the definition of a hybrid controller, such as the event-based hybrid controller by Yang *et al.*^{[17]} or the Maxwell model-based controller proposed by Fu *et al.*^{[18]}. However, the performance achieved by such approaches is not comparable to the ones achieved by the switching logic-based approaches. This paper aims to modify the approach of Ott *et al.*^{[13]} in order to further improve its performance. In particular, considering the approaches in ^{[13-15]}, the inertia parameters of the models considered for the robot simulation were of the order of ^{[14, 15]} moved in the neighborhood of its equilibrium position, *i.e.*, where the inertia matrix *i.e.*, the combined behavior of impedance and admittance controllers). In addition, the improvement of the switching conditions (from impedance to admittance control) can be addressed in order to guarantee the continuity (*i.e.*, the smoothness) of the interaction force. The main goal of the proposed paper is to address these issues.

### 1.3. Paper Contribution

Based on the discussion provided in the previous section, the main goals of the proposed paper are as follows:

(i) Satisfy the continuity of the interaction force in switching from impedance to admittance control when a feedforward velocity term is present.

(ii) Adapt the switching parameters to improve the performance of the hybrid control framework, thus better exploiting the properties of both impedance and admittance controllers.

While Aim (i) is addressed by the definition of proper initialization conditions at the switching time from impedance to admittance control taking into account a feedforward velocity term; Aim (ii) is addressed by adapting the switching parameters on the basis of the resulting robot inertia parameter (for the adaptation of the switching parameter

The implemented methodologies were evaluated in simulation in Matlab, making use of a Franka EMIKA panda robot as a reference robotic platform, comparing the achieved results with respect to the standard method, which uses *et al.*^{[13]}, to demonstrate the improved performance (*i.e.*, improved performance of the combined impedance/admittance controllers during interaction with a variable stiffness environment).

### 1.4. Paper outline

The paper is structured as it follows. Section 2 describes the hybrid controller of Ott *et al.*^{[13]}, together with the low-level controller (*i.e.*, impedance and admittance controllers). Section 3 introduces the modified hybrid control framework in order to deal with the discussed open issues of the state-of-the-art. Section 4 shows the achieved results of the proposed modified hybrid controller with respect to the one using *et al.*^{[13]}. Section 5 states the conclusions of the paper.

## 2. HYBRID IMPEDANCE–ADMITTANCE CONTROLLER

Impedance and admittance controllers are two well-known strategies that are used to assign a specified dynamic behavior to a robot interacting with the surrounding environment. Both controllers aim to impose a target dynamic behavior on the controlled manipulator, as described by the following expression ^{[19]}:

where

While impedance control is exploited in the interaction with stiff environments, admittance control is exploited in the interaction with soft environments (or in the case of free-motion tasks). Hybrid controllers, therefore, aim to combine the performance of both strategies to unify their behaviors into one control strategy, suitable for all interaction conditions.

In the following, the definition of impedance and admittance controllers is recalled, together with the definition of the hybrid controller.

### 2.1. Robot dynamics modeling and control design

To design the impedance and admittance controllers, the robot dynamics can be modeled as follows ^{[19]}:

where *i.e.*, external interaction torques). *i.e.*, *i.e.*,

where *i.e.*, *i.e.*,

### 2.2. Impedance control

The Cartesian impedance control can be defined as it follows ^{[19]}:

where

### 2.3. Admittance control

The admittance control generates a position reference ^{[19]}, slightly modifying the controller in Equation (1):

where

where

### 2.4. Hybrid impedance/admittance control

The hybrid impedance/admittance control can be implemented on the basis of the approach proposed by Ott *et al.*^{[12]}, so that it would be possible to exploit the combined performance of both control schema to deal with a wide range of interaction environment stiffnesses. The proposed hybrid controller continuously switches between impedance and admittance control. For this purpose, the switching period

where

In Equation (11),

While the hybrid impedance/admittance control proposed by Ott *et al.*^{[13]} provides a useful control framework to combine the impedance control and admittance control performance, the following main open issues are still present in the state-of-the-art: (i) the switching law to ensure continuity in the switching from impedance to admittance control does not include a feed-forward velocity term; and (ii) the performance of the hybrid impedance/admittance control framework (*i.e.*, the resulting combined impedance/admittance behavior) can be improved by adapting the switching parameters that were considered constant by Ott *et al.*^{[13]}. In the following section, these open issues are tackled to improve the hybrid impedance/admittance control framework.

## 3. ADAPTIVE SWITCHING PARAMETERS

In this section, two main improvements are proposed for the hybrid impedance/admittance controller proposed in ^{[13]}: (i) proper initialization of

### 3.1. Proper initialization of the admittance controller

To guarantee the continuity of the interaction force while switching from impedance control to admittance control within the hybrid controller, the proper initialization of

#### 3.1.1. Algebraic switching method

To guarantee the continuity of the interaction force while switching from impedance control to admittance control, the following equality has to be satisfied:

By isolating

By differentiating Equation (13), it is possible to compute

#### 3.1.2. Differential switching method

By isolating

By taking its time derivative, it is possible to compute

Equations (15) and (16) can be integrated while impedance control is activated, in order to compute

#### 3.1.3. Iterative switching method

Equation (12) shows that there are infinite combinations of

Iterative switching method algorithm.

Figure 3. Convergence of the proposed iterative switching method considering the error

### 3.2. Adaptive switching logic

The switching logic is defined by the following two parameters: the switching period ^{[13]}, such parameters affect the performance of the hybrid impedance/admittance controller. In particular, based on the equivalent inertia of the controlled robot along the direction *i.e.*, the robot used for the simulation results analysis in Section 4), the equivalent inertia ^{[20]}) varies in a wide range based on the robot configuration. Indeed, to improve the performance of the hybrid controller, such an inertia variation has to be considered for the adaptation of the switching parameters. To support the above discussion, the modification of the achieved hybrid control performance can be seen in Figure Figure 6a, 6b, 6c for the the algebraic, differential, and iterative switching method, respectively, on the basis of the imposed value for the switching parameter ^{[13, 14]}, the duty cycle

Figure 5. Cartesian equivalent inertia

Figure 6. Hybrid control performance on the basis of the switching parameter

In the following, the adaptation of both the switching period

#### 3.2.1. Variable switching period $$ \delta $$

To tune the switching period

where

To tune the *et al.*^{[13]}. By introducing the performance index

where

for a given interaction environment

By simulating the interaction with different environments (*i.e.*, having different stiffness) and values of the duty cycle

To evaluate the cost function *et al.*^{[13]}, having the hybrid controller in interaction with *soft*, *medium*, and *stiff* environments with stiffness values of

Figure 7 shows the cost functions *i.e.*, making it possible to highlight the optimized

Figure 7. Contributions to the cost function for the soft, medium, and stiff environments, considering an inertia parameter

Figure 8. Normalized cost functions for soft, medium, and stiff environments and total normalized cost. The minimum for every value of the inertia

Figure 9.

It has to be noted that the optimized value of *e.g.*, the range of variation for

#### 3.2.2. Variable duty cycle $$ n $$

The adaptation of the duty cycle parameter

This means that, when interacting with a soft environment (*i.e.*, *i.e.*, *et al.*^{[13]} and shown in Figure 10 into the one shown in Figure 11. The adaptation law proposed by Ott *et al.*^{[13]} evaluates

**Remark 1.** The environment stiffness parameter ^{[21, 22]}.

**Remark 2.** The stability of the controller can be addressed following the work in ^{[12, 13]}.

## 4. RESULTS

In this section, the results achieved by the proposed improved hybrid impedance/admittance controller are discussed, comparing them with the standard methodology where ^{[13]}. A simulation study was performed employing a reference robotic platform, a Franka EMIKA panda robot. Matlab was used as a simulation platform, making use of the robot modeled dynamics in ^{[20]}.

### 4.1. Task description

To evaluate the performance of the improved hybrid controller, an interaction task along the vertical Cartesian direction *i.e.*, with a control frequency of

### 4.2. Control parameters definition

The parameters for the impedance controller were chosen as follows:

The diagonal elements of the matrix

The parameters of the admittance controller were imposed as follows:

where

The values used in Equation (17) were imposed as in the following:

### 4.3. Results evaluation

The performance obtained when applying the hybrid control with the *algebraic*, *differential*, and *iterative* switching methods can be seen in Figure 13a, 13c, 13e, respectively, considering fixed and variable ^{[13]} is used, while Figure Figure 16b shows the values obtained with the proposed adaptation strategy in Equation (22). It is possible to observe how *et al.*^{[13]}.

Figure 13. Hybrid control performance comparison considering fixed

Figure 14. Variable

## 5. CONCLUSIONS AND FUTURE WORK

This paper addresses two open issues present in state-of-the-art (*i.e.*, satisfying the continuity of the interaction force in the switching from impedance to admittance control when a feedforward velocity term is present and adapting the switching parameters to improve the performance of the hybrid control framework, better exploiting the properties of both impedance and admittance controllers) to improve the hybrid impedance/admittance control performance in the execution of interaction tasks. The modified methodology's performance (*i.e.*, to verify the improved combined impedance/admittance behavior) was evaluated in simulation, comparing the achieved results with those obtained with the standard method that used *et al.*^{[13]}, making use of a Franka EMIKA panda robot as a reference robotic platform. The obtained results show the improved mixing impedance/admittance performance of the modified hybrid controller.

Future work is devoted to the design of a hybrid impedance/admittance controller exploiting AI techniques to additionally implement motion/force control capabilities (such as in Xu *et al.*^{[23, 24]}). Furthermore, AI techniques will be applied to improve the switching strategy, together with embedding the online estimation of both the robot and the environment modeling into the hybrid control framework.

## DECLARATIONS

### Authors' contributions

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Roveda L, Formenti A

Performed data acquisition, as well as provided administrative, technical, and material support: Formenti A, Roveda L, Shahid AA, Piga D, Bucca G

Paper writing: Roveda L, Formenti A, Shahid AA

Supervision: Piga D, Bucca G

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

The work has been developed within the project ASSASSINN, funded from H2020 CleanSky 2 under grant agreement n. 886977.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

All the authors accept to publish the paper.

### Copyright

© The Author(s) 2022.

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## Cite This Article

## How to Cite

Formenti, A.; Bucca G.; Shahid A. A.; Piga D.; Roveda L. Improved impedance/admittance switching controller for the interaction with a variable stiffness environment. *Complex. Eng. Syst.* **2022**, *2*, 12. http://dx.doi.org/10.20517/ces.2022.16

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