Impedance control of series elastic actuators: Passivity and acceleration-based control

Abstract

Series elastic joints allow force and impedance controllers to be implemented on high torque and high power density motors. Several impedance controllers have been proposed whose stability is usually analyzed by means of passivity-based tools such as the Z-width characterization. This paper proposes an overview of existing impedance control solutions for series elastic joints and derives the passivity characterizations that are still missing in the literature, thus providing a complete and coherent overview of the existing solutions. Within this overview, we highlight the advantages of impedance control based on positive acceleration feedback showing improved stability robustness and impedance accuracy with respect to existing solutions. These advantages are theoretically motivated (considering ideal conditions) and experimentally validated.

Introduction

Series elastic actuators (SEA’s) are an emerging technology to achieve high fidelity force control of high power density motors [1]. In fact, series compliance can dramatically improve explicit force control robustness [2], [3]. Series elastic joints have been successfully applied to humanoid robots (e.g. NASA’s Valkyrie [4], Virginia Tech’s THOR [5] and IIT’s COMAN [6]), quadrupeds (e.g. ETH’ s StarlETH [7]), modern rehabilitation and assistive robotics [8], [9], [10], [11] and cooperative robots (e.g. RethinkRobotics Baxter [12]). Most of these applications are based on impedance control and need to deliver forces with a high level of safety and accuracy. While safety is a primary need for robots that interact with humans or with unstructured environments, the demand for high accuracy is increasing only recently. As an example, this requirement is pushed by novel haptic interfaces which have been proposed in the last years. Among others, Basafa et al. designed a haptic laparoscopic device with three degrees of freedom actuated by SEA’s [13]; Zinn et al. proposed a haptic interface with large workspace based on the Distributed Macro-Mini concept where the macro actuation is given by SEA’s [14]; Oblack et al. proposed a multi-purpose rehabilitation haptic device using series visco-elastic actuators [15]; Parietti et al. designed a haptic device with a series visco-elastic elements for very fine force rendering, in the range of the human sensory accuracy [16]. In these haptic devices, the choice of SEA’s has been usually motivated by the decoupling effect of the series spring which allows to mask the motor inertia and allows the accurate rendering of even very low forces. Differently from traditional haptic interfaces, existing impedance controllers for SEA’s make use of explicit force feedback, meaning that the force is explicitly measured and fed back to the control system. This is because implicit force control (where the force is delivered in open loop by controlling the motor current) cannot mask the motor inertia and cannot damp the series spring oscillations.

The control problem of physical interaction (with humans or with unstructured environments) involving explicit force feedback is considered a hard challenge in robotics. Most of the proposed solutions are based on the passivity interaction paradigm, which provides a high level of stability robustness [17]. In particular passivity of the controlled robot is a sufficient and necessary condition to guarantee a stable interaction with any passive environment [18] including humans who are usually assumed as passive systems [19]. Consequently several passivity-based control (PBC) algorithms have been proposed to shape the output impedance or the output force of SEA’s. The first passive force controller for SEA’s was proposed by Pratt and Williamson [1]. Their solution is based on acceleration feedback that forces the motor to have the same acceleration of the environment, thus compensating for load motion and leading to robust performance, i.e. predictable error dynamics. Quite surprising this important feature was not explicitly highlighted neither in the original work nor in the following literature which has been focused more on stability robustness rather than performance robustness. Another investigation on PBC of SEA’s has been conducted by Vallery et al. where a velocity sourced impedance control schema was considered [20]. They found out that “SEA cannot display a higher pure stiffness than the spring stiffness if passivity is desired”. Thus the spring design cannot be arbitrarily compliant but must be tailored to the maximum desired stiffness leading to a trade-off between force control robustness (which requires compliance) and the maximum displayable stiffness.

One of the outcomes of the research described in this paper is to show that the same limitation highlighted by Vallery et al. holds for several existing control architectures: admittance control [21], impedance control [22] and parallel force-position control [23]. The admittance control architecture was proposed by Pratt et al. to reduce the force control bandwidth requirement in high impedance rendering [21]. We will show that even this algorithm cannot passively render a stiffness higher than the stiffness of the physical spring. A similar outcome emerges by analyzing the passive impedance control of the DLR lightweight arm where a parallel force-position architecture is implemented [23].

In conclusion, we will show that at the current state of the art there exist no passive algorithms that allow to overcome the physical spring stiffness. Further limitations arise when a virtual Voigt model impedance is desired, i.e. the parallel of a spring and a damper. For example an impedance controlled SEA with velocity controlled motor cannot passively display a pure Voigt model [24] and we will show that the same result holds for other control architectures. Thus, the first contribution of this work is to derive missing passivity conditions for existing impedance control architectures providing a coherent framework of passivity results.

The second contribution is a novel algorithm that allows in theory to passively overcome the physical spring stiffness and to passively display a pure Voigt model. This algorithm is inspired by the seminal work by Pratt and Williamson [1] who used load acceleration feedback to control the SEA output force and to cancel out the influence of load dynamics in force control. We will formally show that by taking advantage of load dynamics cancellation it is possible to passively render any passive impedance. The intuition behind this approach is that by exactly compensating for the load motion, load (or environment) uncertainties disappear. In particular we refer to (perfect) load motion compensation as the ability to move the motor homokinetically with the load or the environment. Thus load motion compensation is a way to obtain virtual backdrivability, i.e. to backdrive a non-backdrivable motor by control. Indeed, the force to accelerate the motor is transferred to the motor input by control and it is used to compensate for the motor inertia. In the past, the concept of load motion compensation has been already discussed in the force control literature and a generic framework of solutions is presented in [25]. However, the effects on impedance rendering and passivity have never been analyzed. Other examples of SEA’s force control where the load dynamics is explicitly taken into account includes disturbance observer architectures [4], [26], [27], [28], adaptive [29], [30], robust [31] and sliding-mode [32] controllers. However, none of these works investigates the effect of load motion compensation on impedance rendering. We highlight that the terms “load” and “environment” can be often considered as equivalent: they both refer to the dynamics the SEA or the robot is in contact with. A typical case is physical human-robot interaction (pHRI) where the environment include or is identified with the human.

To address the issues described above, the paper is organized as follows. To introduce the reader, Section 2 summarizes existing impedance control algorithms for SEA’s. Section 3 derives passivity conditions that are currently missing in the literature and provides a summary and easy-to-compare view of passivity results. Section 4 proposes a novel impedance control algorithm based on positive acceleration feedback. Section 5 experimentally compares the existing algorithms to our solution from the point of view of stability robustness and impedance accuracy. Finally, conclusions are drawn in Section 6.