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There are two main categories of force control schemes: hybrid position-force control and impedance control. However, the former does not take into account the dynamic interaction between the robot’s end effector and the environment. In contrast, impedance control includes regulation and stabilization of robot motion by creating a mathematical relationship between the interaction forces and the reference trajectories. It involves an energetic pair of a flow and an effort, instead of controlling a single position or a force. A mass-spring-damper impedance filter is generally used for safe interaction purposes. Tuning the parameters of the impedance filter is important and, if an unsuitable strategy is used, this can lead to unstable contact. Humans, however, have exceptionally effective control systems with advanced biological actuators. An individual can manipulate muscle stiffness to comply with the interaction forces. Accordingly, the parameters of the impedance filter should be time varying rather than value constant in order to match human behavior during interaction tasks. Therefore, this paper presents an overview of impedance control strategies including standard and extended control schemes. Standard controllers cover impedance and admittance architectures. Extended control schemes include admittance control with force tracking, variable impedance control, and impedance control of flexible joints. The categories of impedance control and their features and limitations are well introduced. Attention is paid to variable impedance control while considering the possible control schemes, the performance, stability, and the integration of constant compliant elements with the host robot.

When a robot is in contact with the environment via its end effector, some important points should be noted:

Given a specific degree of freedom, it is not possible to independently regulate the position and the contact force. For example, if the task of the target robot is to write something, neglecting control of the interaction force may lead to either loss of contact or hard pressure on the target environment [

In addition, the robot loses some degrees of freedom (DoFs) during the contact phase. Consequently, the generalized coordinates of the target robot might be larger than its DoFs due to its constrained motion; this constitutes a closed-chain mechanism with redundant coordinates [

The robot may change its configuration during a transition from an open-chain mechanism to a closed-chain mechanism. In effect, three motion phases can be produced: the free motion phase, the contact motion phase (impact phase), and the constrained motion phase. Each phase can have its own features and control law [

One of the solutions to regulate and control the interaction forces is hybrid position/force control proposed by Raibert and Craig [

In effect, impedance control plays an important role in any workspace that involves human-robot interactions. The idea behind it is to control the mechanical impedance of a host robot regulating the interaction forces produced by the coupling between the robot and the environment dynamics; mechanical impedance can be defined as the ratio of the output force to the input velocity (motion). For linear systems, mechanical admittance is the inverse of mechanical impedance; it can be defined as the ratio of input velocity (motion) to the output force. In general, the robot can ideally behave as an impedance and the contact environment is an admittance; however, this could not be the case for multibody robotic systems with heavy links and actuators [

In view of the above, this paper is aimed at summarizing the different schemes of impedance control with miscellaneous control modes (see Figure

A general classification of the impedance control approaches. The paper is organized according to the depicted classification.

The paper is organized as follows: Section

Although the impedance control schemes are referred to as an indirect force method, some of these schemes can include a force tracking loop with the impedance target, e.g., the position-/velocity-based impedance control (admittance control) can be modified to improve the interaction force tracking problems (this is discussed in the following sections). As mentioned, the idea of impedance or admittance control is to generate a dynamic relationship between the interaction force/torque and the position/velocity trajectory of the robot end effector by using the virtual

Description of impedance control for a robot in contact with the external environment [

By tuning the parameters of the impedance parameters, a suitable performance can be obtained for the host robot; there is a deviation in robot motion associated with and coupled with the deviation of interaction force. Basically, impedance control may consist of two nested control loops: an outer impedance control loop and an inner position/velocity/force control loop. For more details on the interaction force control schemes, see [

The three related subcategories of impedance control are stiffness control [

Target impedance dynamics (outer impedance loop) is preferably expressed in terms of the task coordinate frames, since the task geometry may decide which directions are motion constrained and force sensitive [

Consider an ^{nd} Lagrangian formulation can be used for modeling. Thus, the dynamic equation of rigid joint fully actuated robots can be expressed in joint space as follows.

In general, the following points should be noted:

For a robot having

In general, there are two possible aspects of redundancy problems, i.e., motion redundancy and torque redundancy [

Equation (

Consideration of actuator dynamics is important for a robot with high-velocity movement and highly varying loads. For more details on the effect of neglecting actuator dynamics, the reader is referred to [

On the other hand, (

Section

The idea behind force-based impedance control (simply called “impedance control” in the literature) is to make the controller react to the motion deviation by generating forces [

Generic diagram of force-/torque-based impedance control [

To motivate the concept of impedance control, consider the following simple second-order system (Figure

System dynamics in contact with the external environment [

As stated, impedance control attempts to make a dynamic relationship between the interaction force and position error by assuming a virtual mass-spring-damper model with the desired trajectory; accordingly, the target impedance function can be expressed as [

Changing the structure of the target impedance dynamics or the behavior of the target impedance coefficients leads to different impedance control strategies. Substituting (

As can be seen, the feedback controller of (

Consider the case

In effect, there are three possible models for representing the target impedance dynamics that correlate the dynamic relationship between the position and contact forces

Equations (

Seraji and Colbaugh [

In admittance control, the controller aims to soften the stiff position source via reacting to the interaction forces by imposing deviation from the desired motion [

Schematic diagram of position-based impedance control [

Below is a simple motivating example that describes the position-based impedance control. However, for the velocity-based impedance control, a similar strategy can be used by replacing the desired and commanded position references with the velocity reference.

For the position-based impedance control of the previous 2^{nd}-order system described in (

System dynamics in contact with the external environment. The philosophy of impedance target dynamics changes due to adding the commanded impedance trajectory

The inner position control can be implemented using the proportional-integral-derivative (PID) family in our simple example; thus, the control law can be expressed as

In effect, the well-known nonlinear schemes, e.g., feedback linearization control (computed torque control), passivity-based control, robust sliding mode control, and mode reference adaptive control, can be used for the inner position control loops [

In effect, force-based impedance control and position-/velocity-based impedance control are based on the assumption of a force-controlled system and a position-controlled system; therefore, their performance and stabilities may differ [

For the force-/torque-based impedance control, an inner feedback loop for force/torque is optional while for position-based impedance control, the inner position loop is required

Since most of the industrial electromechanical manipulators are equipped with servo position control loops, position-/velocity-based impedance control might avoid redesigning the inner position loop

For the desired stiff impedance behavior, force-based impedance control may encounter instability problems due to the amplification of noise. If the environment is soft (compliant), the stiffness of the end effector should be stiffer and vice versa. Accordingly, force-based impedance control might be suitable for interaction with a stiff environment. In contrast, position-based impedance control is more suitable to implement stiff behavior than compliant behavior, i.e., it is suitable for interaction with a compliant environment

The performance and stability of force-based impedance control may depend on back drivability and the amount of friction for the host system, whereas the performance of position-based impedance control can depend on the performance of inner position control and the quality of the force measurement

For more details on the differences among these categories of impedance control, see [

Literature proves that an inner velocity control loop can improve the performance and the stability problems associated with impedance control. However, the following points should be considered:

Using a force loop around a position loop seems to be very natural and, therefore, this was exactly the mainstream approach used in the 1980s; see [

Referring to (

In some robotic applications, the desired position trajectory can be unknown, and thereby, the use of an inner velocity control loop is more suitable. Examples of these applications are the unknown final destination of human-robot cooperation [

In humans, the stiffness of muscles plays an important role in dextrous and robust motions. For example, the human arm can control the interaction contact force by modifying its muscle stiffness such that the interaction contact force can be either increased by making the arm stiffer or decreased by reducing the arm’s stiffness. In addition, an individual can keep the force tracking error within a specified range in the presence of disturbances and uncertainty [

In general, most robotic systems need to be in contact with the external environment. Regulation of the interaction force is necessary to avoid problems related to instability and safety. Some robot applications include control and stabilization of the constant value interaction force, such as with deburring, welding, and grinding [

Accordingly, the exact position and force tracking may not occur in conventional impedance control strategies. The main limitation of impedance control is that the interaction forces are controlled indirectly by selecting the desired impedance dynamics. However, this may demand accurate knowledge of environment parameters (e.g., environment location and stiffness) which are difficult to specify in practical applications [

To illustrate the importance of knowledge of the environment parameters, consider the following scalar target impedance function

If the desired reference trajectory maintains constant values, their first and second derivatives are equal to zero. Thus, (

Using a simple spring model to represent the deformation of the environment (assuming that environment stiffness dominates its deformation), the interaction force can be expressed as

Rewriting the above equation to get the end effector position leads to

Inserting the force error in (

Because the objective of the inner position control loop is to track the commanded compliant impedance references

Substituting (

If the impedance system reaches the steady-state region assuming that the desired environment force is of constant value, the steady-state force error can be expressed as

Let

Accordingly, the convergence of interaction force tracking could not be ensured in position-based impedance control, especially with uncertain environment stiffness and uncertain modeling of the host robotic system [

Seraji and Colbaugh [

Kim et al. [

For more details on force tracking-based admittance control, see [

In effect, making the system stiffness variable imitate human behavior is considered to be variable impedance control (see explanation in the following section)

The knowledge of environment stiffness and location is necessary for force tracking-based admittance control

An inner velocity control loop can alternatively be used with features discussed in Section

The derivative of the environment force error is required in some schemes, that is, undesirable. The two possible techniques to solve this problem are (i) making a filter for the sensed force signal then differentiating the filtered signal [

Most researchers assume that the decoupled outer impedance filter can simplify the control problem such that stability analysis and performance of the proposed force tracking impedance filter may depend on the linear control theory, such as root locus analysis and Routh-Hurwitz stability

An important observation is that virtual stiffness of impedance behavior can lead to steady-state errors; therefore, cancelling this term may lead to zero steady-state errors [

For most biological movements, muscles behave as mechanical actuators with a nonlinear stiffness behavior; according to biological studies, muscle viscosity can be considered constant. The force-velocity relationship includes nonlinear characteristics during contraction and stretching; increasing the applied force may result in an increase in muscle stiffness. It is important to note that the slopes of the impedance curve represent the muscle impedance associated with muscle movement [

There are essential applications that the robot is in contact with the human such as exoskeletons, orthosis, and prostheses. In view of the above statement, using the conventional impedance control with fixed coefficients, e.g., fixed stiffness, cannot achieve the required target impedance for the human-robot interaction. Accordingly, variable stiffness-based impedance control can improve the performance of the desired force tracking and the dexterity of the robotic system. It is a suitable strategy for modulation of the parameters of the impedance behavior such that stability is guaranteed and the performance is improved and safer. This policy of changing stiffness is explained above (Section

Mathematically, the target impedance behavior with variable parameters can be expressed as

In view of the above statement, there are two main objectives for equipping the target impedance with variable impedance:

to track interaction force references; please see Section

to increase adaptability and to imitate the biological behavior during contact with different environment stiffness

However, straightforward implementation of impedance control with time-varying virtual impedance parameters can destroy passivity conditions of the system unless a proper impedance model is selected. To prove this, consider the 2^{nd}-order dynamics system combined with the target impedance model described in (

Taking the derivative of the last equation and substituting (

According to the last equation, the time-varying virtual stiffness could violate the passivity condition, whereas the virtual damping term could have a positive effect on the energy dissipation. However, assuming constant-value virtual parameters can ensure the system stability as follows.

Integrating the last equation to get the following satisfied passivity condition gets

See [

It is important to keep in mind that the time-varying virtual stiffness parameter can be the critical determinant of the system stability. Literature proves that there are two options for time-varying virtual impedance mass/inertia: (1) it can be of constant value with no effect of Lyapunov stability or (2) it can be of a value equal to the mass/inertia of the robot end effector. The last case can be exploited to design a control law free of contact force feedback; see [

In effect, four techniques are possible to deal with active variable impedance control:

manipulation of the virtual stiffness term such that it is related to interaction force error via a PID family controller; see the work of [

neglecting the virtual stiffness term of the impedance model and manipulating the virtual mass and damping terms. Tsumugiwa et al. [

augmentation of the impedance model with an energy-storing element whose role is to store the energy dissipated by the controlled system such that the passivity conditions are satisfied. With this scheme, impedance control with time-varying stiffness matrices can be a powerful tool to deal with a compliant environment that requires time-varying interaction forces. This technique has been called energy tank-based impedance control and implemented by [

design of adaptive laws for tracking the virtual damper and spring parameters. However, this technique could impose constraints on the values of the virtual damping and stiffness in order to ensure the system stability; for more details, see the work of [

However, some works injected the time-varying stiffness directly to the impedance model without considering the overall system stability and the associated passivity conditions, e.g., see [

There are two strategies for imitation of human impedance behavior: variable impedance actuators [

A challenge in the application of variable impedance target with human behavior is how to transfer the impedance characteristics from humans to robots and ensure overall system stability. Various techniques are available for estimation of human impedance: most are based on neurological schemes, such as the human central nervous system [

In general, if the robot is to be freely driven by the human, robot impedance should be low; zero stiffness is recommended in this case. For fast motion purposes, virtual robot damping should be decreased and vice versa, whereas decreasing virtual inertia may lead to instability problems [

On the other hand, regulation of the virtual stiffness of the host robot is necessary for surgery, rehabilitation applications, and collaborative robots [

With constant parameter-based impedance control, the passivity property is conserved; however, with arbitrary time-varying parameters for impedance behavior, the passivity property can be lost [

In this section, impedance control of robots with flexible elements is discussed. The focus is on constant impedance series elastic actuators (SEAs). The cascade control combined with an outer impedance loop is often proposed for these types of compliant actuators; for more details on cascade control theory, see [

Simplified schematic for series elastic actuators (SEA) [

Accordingly, the actuator is called constant SEA or variable stiffness actuator (VSA) based on the behavior of the designed joint stiffness (constant stiffness or variable). In this category of actuators, the actuator does not control the link directly but will exchange energy with the transmission system that generates the flexible torque that actuates the link. In recent technology, flexible joints are integrated with robots to guarantee safe motion during the contact phase or to attenuate the impact shock of unexpected forces [

It serves as an accurate torque source and as a low-cost torque sensor

The elastic element also serves as a compliant interface between the human and the robot, protecting the user and actuating system from sudden shocks and improving back drivability characteristics. In effect, the contact force can be indirectly regulated and controlled by the passive elements

The motor is isolated from shock loads, and hence, the dynamic effects of backlash and friction can be filtered by the flexible element

A drawback is the reduced large torque bandwidth due to motor saturation

The following points need to be considered when designing flexible joints:

The output flexible torque is important in the performance of interaction tasks; the flexible element should be exploited in the control structure rather than dealing with it as a disturbance source

The flexible element (e.g., spring) acts as a force sensor allowing the actuator output force to be controlled, and hence, the design of the control law could be easy; see [

The behavior of the flexible transmission cannot be known completely for variable impedance actuators, due to the inherent nonlinearity and associated complexity

Recalling (

According to (

Different techniques are available to deal with the control of flexible joints: decoupling control schemes [

The rest of this section considers specifically the possible problems associated with nested control loops of the SEA. Many researchers used simple force control of SEA-driven robots based on linear control theory. However, the system stability and passivity conditions should be satisfied in order to achieve feasible performance [

Multilevel control of the LOPES robotic system where the subscript

Tagliamonte et al. [

Proposed cascade control with three control loops: high-level stiffness servo loop, inner embedded damping servo loop, and the innermost torque control [

Mosadeghzad et al. [

The discrete impedance control system may require the lower bounds for the bandwidth of the inner control loop; however, in a continuous time control system, larger values for the inner loop gains can be obtained to ensure the stability and achieve high overall bandwidth

With the inner torque control loop, the model of the host robot should be known to avoid instability. However, using only an inner torque loop, the system can drift and this is because torque control only stabilizes the torque response of the system but does not provide internal stability. This can be overcome by setting an outer loop with PD control, a zero reference position, and velocity and even setting the

Li et al. [

In view of the above statement, the following points can be noted:

The control architecture of the SEA can consist of three nested control loops: an innermost velocity loop, an intermediate torque loop, and the outer impedance control loop that renders virtual impedance for safe and comfortable human-robot interactions: see [

A powerful tool for tuning the gains of cascade control of the SEA and determining the ranges of the virtual impedance target is the passivity theory. Most work has focused on stability/passivity constraints of cascade control for single SEA-actuated joints using linear control theory. Extending the work for MIMO robots considering the nonlinearities, time delay problems, and stability problems associated with the coupling nested loops is not straightforward; see [

For pure virtual spring impedance target, the maximum value of the virtual spring impedance can overcome the physical stiffness while retaining passivity [

Most of the standard control schemes of soft robots such as high-gain robust/adaptive control, feedback linearization, and active impedance control attempt to regulate/control the target system at the expense of stiffening it. Therefore, Santina et al. [

The constant impedance actuators described above may have limitations associated with dealing with the different tasks and motion frequencies; the different tasks need variable stiffness (impedance) actions that could be lost in the SEAs. Therefore, robotic systems with VSAs are capable of rejecting disturbances, storing energy, and controlling the end effector stiffness in contact space [

In general, there are three control schemes of VSAs: (i) simultaneous control of position and stiffness control [

The proposed control schemes, the performance, and stability have not yet been extensively investigated. The stability of impedance control associated with VSAs requires more research. However, impedance control associated with inner torque control is the easiest control scheme to deal with constant and VSAs. For more details on the control architecture and stability of VSAs, see [

This paper is aimed at systematically introducing the features and limitations of the categories of impedance control schemes. Basically, impedance control can be classified as force-based impedance control and position-based impedance control. The conventional impedance control schemes do not consider the force tracking problems in the outer impedance filter, resulting in a deviation of the desired force references. Accordingly, modification of the impedance filter to satisfy the force-tracking problem is a motivating technique of imitation of human behavior. As mentioned, one strategy for force tracking-based impedance control is to change the virtual stiffness. Therefore, a clear connection is required between variable impedance control and force tracking-based impedance control. On the other hand, changing the impedance parameters is not trivial; investigation of the stability problems of variable impedance control requires additional work.

Impedance control of flexible-joint actuated robots remains a challenge. Control of robots with constant impedance joints could be easier than variable impedance joints. In variable impedance actuators, the stiffness is an added variable output that should carefully be controlled. In general, an outer impedance filter integrated with torque control is an effective strategy to solve this category of transmission. In general, a careful control architecture is required to exploit joint flexibility. For example, using the standard feedback control schemes may make the system stiffer, and hence, the system behavior changes. Therefore, bioinspired-based control systems such as feedforward action can work well to exploit the system impedance.

The authors declare that they have no conflict of interest.

This work was funded by the Key Project of the National Natural Science Foundation of China under award no. 61233014, by the National Natural Science Foundation of China Project under award no. U150920072, by the Science and Technology Innovation Major Project of Shandong Province, China, under award no. 2017CXGC0903, and by the Key Research and Development Project of Shandong Province under award no. 2016ZDJS02A07.