High-Bandwidth Active Impedance Control of the Proprioceptive Actuator Design in Dynamic Compliant Robotics

Abstract

Dynamic compliant robotics is a fast growing field because of its ability to widen the scope of robotics. The reason for this is that compliant mechanisms may ensure safe/compliant interactions between a robot and an external element—for instance, a human operator. Active impedance control may widen the scope even further in relation to passive elements, but it requires high-bandwidth robust torque and active impedance control which induces high-noise issues even if high-end sensors are used. To address these issues, a complete controller design scheme, including Field-Oriented Control (FOC) of a Brushless Direct Current (BLDC) motor, is proposed. In this paper, controller designs for controlling the virtual impedance, motor torque and field are proposed which enables high-bandwidth robust control. Additionally, a novel speed and angle observer is proposed that aims to reduce noise arising in the angle sensor (typically a 12-bit magnetic encoder) and a Kalman/Luenberger based torque observer is proposed that aims to reduce noise arising in the phase current sensors. Through experimental tests, the combination of the controller designs and observers facilitated a closed-loop torque bandwidth of $2.6$ $\mathrm{k}$$\mathrm{Hz}$ and a noise reduction of $13.5$ dB (in relation to no observers), at a sample rate and Pulse Width Modulation (PWM) frequency of 25 $\mathrm{k}$$\mathrm{Hz}$. Additionally, experiments verified a precise and high performing controller scheme both during impacts and at a variety of different virtual compliance characteristics.

1. Introduction

Compliance enables robotics to be used in contexts that include transitions of energy between a robot and fragile or unknown external elements—for instance, interactions with humans or walking on rugged terrain. This ability widens the scope of robotics which makes compliant robotics a fast advancing field. For this reason, the advancements in compliant robotics has made its footprint in areas where compliant interactions are required—for instance, legged robotics [1], rehabilitation robotics [2], wearable robotics [3], robotics surgery [4], robotic prostheses [5], soft robotics [6], collaborative robotics [7] and active vehicle suspension [8].

1.1. Compliant Actuators

In order to facilitate compliance in a robot, it must include a compliant actuator. Five of the most commonly used electromagnetic actuator designs for compliant actuation of robots [9] are illustrated in Figure 1.

Conventionally, compliant robots are equipped with passive springs and dampers (e.g., SEA and PEA) which enable a smooth transition of energy between the robot and an external element without damaging either and ensuring stable operation. However, the dynamic response is restricted to those enabled by the selected passive elements which limits the dynamic response and thereby the variety of achievable interactions. It is often required to adapt the impedance (in order to retain stable operation) which can be done by deploying the VSA or an active impedance controller (often referred to as virtual compliance) [10,15,16] which is obtainable using GMS or the proprioceptive actuator.

The high gear ratios used in GMS, SEA, PEA and VSA enable high torque. However, as the transmission path from the motor shaft to the end-effector of GMS, SEA, PEA and VSA includes many components (gears, sensors, springs, etc.) that introduce elasticity and inertia, the force transparency is relatively low [1,9,12,13]. Hence, these four actuators are not able to accomplish high-bandwidth open-loop force control of the end-effector which is required in order to perform dynamic physical interactions. As the proprioceptive actuator only includes a low ratio gearing and no other mechanical components are required, it is capable of achieving a much higher open-loop force controller bandwidth of the end-effector which enables dynamic physical interactions [1]. In fact, studies show that the dynamic range of achievable virtual impedances (Z-width) mainly depends on the closed-loop torque bandwidth [10]. Using the proprioceptive actuator design, lower gear-ratio can be compensated for by using a high-diameter motor.

In order to enable a wide Z-width, MIT initially designed their Mini Cheetah robot with a closed-loop torque bandwidth of $4.5$ $\mathrm{k}$$\mathrm{Hz}$ [12] by using the proprioceptive actuator design. However, due to the high bandwidth, high audible noise was observed by the author of [12]. This is expected since active impedance control typically involves computing a torque reference based on the error between desired and actual speed and position. As speed needs to be computed at high rates (typically 5 $\mathrm{k}$$\mathrm{Hz}$ to 50 $\mathrm{k}$$\mathrm{Hz}$) from the angular position which is typically measured by a magnetic encoder (typically around 12-bit resolution), noise will be injected into the torque loop. A high closed-loop torque controller bandwidth induces minimal filtering effect in the torque controller meaning that the noise arising from computing the derivative of the angle gets directly injected into the phase voltages and will appear as audible noise in the motor. Additionally, the noise from the encoder and the phase current sensors are being injected into the phase voltages when using Field Oriented Control (FOC) as it is used in the computation of the torque and field components [17].

1.2. Existing Impedance Control Systems

When a robot is deployed in an environment with unknown stiffness and interaction tasks are required (such as human-robot interactions), problems of instability or high force-overshoot may occur which can potentially create a dangerous situation. To solve that issue, a method based on the energy tank theory was proposed in [18] with the purpose of preserving the passivity of the controlled system. To reduce instabilities and force overshoot, [19] uses optimal control by employing a Linear-Quadratic Regulator (LQR) to adapt the controller gains. In addition, an Extended Kalman Filter (EKF) was used to estimate the stiffness of the environment while online.

The concept of collaborative robots (often referred to as cobots) was invented back in 1999 partly to assist people in heavy duty tasks. However, until 2009, the only heavy duty human-assisting tools deployed in the industry were weight compensators/balancers which were not sufficient to avoid back injuries. Therefore, more advanced solutions were needed. Reference [20] proposed a framework that allows for a direct human-machine collaboration (with no minimum security distance) where the robot alternates between active and passive collaboration during assembly with the purpose of unburdening the worker. Reference [21] proposed a manual guidance system that allows for human guidance (by direct contact) of a cobot which acts as an active weight compensator.

In order to increase flexibility of industrial robots, they are often deployed on a mobile base. This adds elasticity to the robot base which affects the dynamics of the interaction between the robot and the environment. Reference [22] proposed an interaction controller that is able to adapt its parameters online to compensate for robot base elasticity.

1.3. Work Contribution

This paper analyzes how the motor is affected by noise arising from the angle sensor and the three phase current sensors. Based on the results quantified by this analysis, a novel angle and speed observer is proposed. Previously proposed observers are mostly used for high-speed sensorless control [23,24,25] rather than for reducing the noise (or increasing the resolution) of a deployed angle sensor at variable speed with minimal phase lag which is needed in dynamic compliant robotics. The proposed angle and speed observer enables filtering as well as precise estimation of mechanical angle, electrical angle and speed for variable speed, high-bandwidth and high-torque control. Additionally, it features minimal phase delay and is intended for FOC commutation of BLDC motors.

Most papers published within the area of active impedance control of compliant robotics (using the proprioceptive actuator design) deal with abstract issues such as the robotics control and mechanical design rather than how to properly design the two main required loops (the torque and impedance loop) for achieving high-bandwidth active impedance control. Therefore, model-based controller design methodologies are proposed which enable high-bandwidth and robust control.

The main contribution of this work is the proposal of a complete and reproducible controller design scheme which facilitates a combination of high-bandwidth (or high Z-width), high-precision and low-noise active impedance control. This mainly consists of the following three parts:

• The torque and field controller design (Section 3.1) describes a “best practice” tuning of the torque and field controller parameters for high-bandwidth and highly stable closed-loop torque control of Brushless Direct Current (BLDC) motors tailored for active impedance controlled compliant robotics.
• The active impedance controller design (Section 3.2) describes novel equations to derive controller gains that ensure a virtual compliance response closely related to the response of its physical counterpart (a mass-spring-damper system).
• The observer designs (Section 6 and Section 7) describe two observers that enable high-bandwidth low-noise motor control. In particular, Section 6 describes a novel observer that is tailored for robust high-bandwidth, low-noise compliant robotics to achieve noise-reduced angle and speed estimations as compared to using the raw angle and speed obtained from the encoder directly.

The novelty of parts 1 and 2 is not significant alone, but they are important parts of achieving the combination high-bandwidth (or high Z-width), high-precision, low-noise active impedance control and are therefore—as a union (including all three parts)—considered novel. Parts 1 and 2 stand in contrast to most papers within active impedance controlled robotics that tend to neglect the low-level control design or use sub-optimal solutions. In addition, in contrast to most papers within high bandwidth active impedance control, this work deals with the problem of noise (arising due to the combination of high bandwidth torque and active impedance control) with the introduction of the novel speed and angle observer. Applying this observer, a high torque bandwidth (comparable to that in MIT’s Mini Cheetah) with low noise issues is obtainable.

Additionally, a novel and reproducible test configuration designed to perform a variety of different tests is proposed. This includes realistic locomotion tests, active impedance tests, collision tests, elevation tests and impact force tests—e.g., to simulate an impact between a legged robot and the ground when dropped from a specific distance or a collision between an industrial or collaborative robot and an external element.

2. Proposed Active Impedance Controller System

The motor controller system covered in this article is based on the Field Oriented Control (FOC) [17] commutation strategy further described in Appendix I. FOC enables high-efficiency, high-bandwidth control using Proportional–Integral–Derivative (PID) controllers. The system includes a plain cascaded control system architecture and includes PD and PI controllers to accomplish high-bandwidth and robust impedance, torque and field control. A complete system diagram is shown in Figure 2.

The outer mechanical angle (${\theta }_{\mathrm{m}}$) controller loop consists of a gain scheduled PD controller which enables active impedance control (virtual compliance) further described in Section 3.2. Within that loop, there is a torque loop that uses a PI controller and the quadrature current ($i_{\mathrm{q}}$) as feedback to ensure a certain torque response. This is further described in Section 3.1. A field loop that uses a PI controller and the direct current ($i_{\mathrm{d}}$) as feedback ensures that no field is generated in the stator windings which ensures maximum torque/current is delivered [26]. A novel and a Luenberger/Kalman observer filter the quadrature $i_{\mathrm{q}}$ and direct currents $i_{\mathrm{d}}$ as well as the rotor angle ${\theta }_{\mathrm{m}}$ to enable low-noise while ensuring that minimal phase delay and high-bandwidth are obtainable.

3. Motor Controller Designs

A cascaded controller architecture is used to enable active impedance control. All three loops (field, torque and impedance) are presented in the block diagram (Figure 3).

A model-based design method is proposed to design the proportional and integrator gains in the PI controllers within the field and torque loops (see Figure 3). It is based on the Nyquist frequency method and the focus is to enable high-bandwidth robust torque and field control. Another model-based method is presented to design the proportional and derivative gains in the PD controller within the impedance loop. This method is based on the pole-placement method and the parameters are designed to accomplish a dynamic response which reflects the response of a desired second order spring-damper system.

3.1. Torque and Field Controller Design

The torque and field controllers are used to control the torque demanded from the higher level controller and the electromagnetic field generated across the stator windings. In this section, a controller design is proposed that is based on the standard and well-known Nyquist method and bilinear transformation as well as best practice design choices. The main purpose of this design approach is to enable high-bandwidth, low-noise, robust control. As depicted in Figure 3, the torque loop is coupled to the motor’s back-Electromotive-Force (back-EMF) (back-EMF coupling term depicted in Figure 3) which creates a low frequency zero in the torque controller loop. This means that even though an integrator is applied in the controller, zero steady state cannot be obtained (assuming ${\omega }_{\mathrm{m}}\ne 0$ rad/s). However, as the proprioceptive actuator design in compliant robotics typically involves low Revolutions Per Minute (RPM) and high torque motor drive, the back-EMF will most often be fairly small, generating a small error. In addition, the high torque controller bandwidth as well as the outer compliance loop will reject this error even further. Thus, a feedforward decoupling compensation (as described in [27]) will generate more issues, especially in terms of noise, than it will benefit for this specific application. The same logic applies to not using a feedforward decoupling compensation to decouple the field and torque loops (qd coupling terms, depicted in Figure 3).

The controller pseudo algorithms used for controlling motor torque and field are shown in (Appendix C). The decoupling terms should be left out of (A15) and (A16) if high-bandwidth (BW > 1 $\mathrm{k}$$\mathrm{Hz}$) is required. The design method proposed in this paper is based on the Nyquist frequency method in which the trade-off between bandwidth and robustness can be selected by a desired amount of phase margin from which the parameters are designed. Designing for 60${}^{\circ }$ of phase margin is proposed as it creates an unconditionally stable system while achieving high-bandwidth. Equations (A24) and (A25) are derived from the bilinear transformed motor model including sampling and delays. The derivation is shown in Appendix G. Note that the design procedure assumes that sensor delays, propagation delays, etc. are negligible. First, the zero of the integrator is designed such that it cancels out the pole of the electrical motor model $-{\tau }_{\mathrm{I}}=-\frac{R}{L}$ which leads to approximately $-90$${}^{\circ }$ phase shift in the entire frequency spectrum (when leaving phase shift based on the low frequency zero and sampling out of the model). This way, the bandwidth is entirely restricted by the sampling and PWM delay and high-bandwidth can be achieved. The desired crossover frequency is determined by solving for ${\omega }_{\mathrm{c}}^{*}$ in Equation (A24) and then solving Equation (A25).

3.2. Active Impedance Controller Design

The active impedance controller is used to compliantly control the angle of the rotor or the trajectory of the end-effector of a robot. As no specific application is focused on in this paper, a simple design methodology is proposed (based on the standard and well-known pole-placement method) that only involves controlling the compliance at the rotor shaft. For example, in a legged robot this could correspond with the hip compliance shown in Figure 4a.

For applications such as legged robots and collaborative robots, it is desired to control the compliance between the end-effector and the actuator (leg compliance in Figure 4a). For these applications, the forward kinematics and polar coordinate transformation described in [28] should be included in the model. The impedance obtained by the controller should be programmable in a way that enables the robot or the user to change the impedance according to a desired model which can be expressed by one of the following equations:

$$\begin{array}{ccc}\hfill \frac{X_{\mathrm{s}}\left(s\right)}{F_{\mathrm{s}}\left(s\right)}=\frac{1}{M_{\mathrm{s}}{s}^2+B_{\mathrm{s}}s+K_{\mathrm{s}}}& \hfill & \hfill \frac{Y_{\mathrm{s}}\left(s\right)}{U_{\mathrm{s}}\left(s\right)}=\frac{1}{\frac{1}{{\omega }_{\mathrm{n}}^2}{s}^2+2\zeta \frac{1}{{\omega }_{\mathrm{n}}}+1}\end{array}$$

(1)

$M_{\mathrm{s}}$, $B_{\mathrm{s}}$ and $K_{\mathrm{s}}$ represent the mass, desired damping constant and desired spring constant, respectively. ${\omega }_{\mathrm{n}}$ and $\zeta$ are the desired natural frequency and damping factor, respectively. Either of the equations in (1) can be chosen as the reference dynamic model depending on the specific application. To be able to simplify the active impedance controller design, the impedance loop in Figure 3 is simplified to the static external load closed-loop system shown in Figure 4b. The external load torque $T_{\mathrm{l}}$ is treated as the input to this closed-loop system while the mechanical motor model is the forward path. The feedback is the product of the PD controller and the torque constant. The inner torque loop is assumed to have a much higher bandwidth than the active impedance controller meaning that it can be removed from the model. In addition, the sampling frequency is assumed to be much higher than the active impedance controller bandwidth thus the active impedance controller is designed as an analogue controller. A static situation is assumed where the desired angle is constantly zero. It is acknowledged that this is not a realistic case. However, the essence of compliance is that it should be able to absorb any external force acting on the system—for instance, a person pulling on a spring acts as an external force and the spring generates force to oppose that. Using Mason’s rule, the closed-loop active impedance controller transfer function is derived:

$$\frac{{\theta }_{\mathrm{m}}\left(s\right)}{-T_{\mathrm{l}}\left(s\right)}=\frac{\frac{1}{Js+B}\frac{1}{s}}{1+\frac{1}{Js+B}\frac{1}{s}({\tau }_{\mathrm{d}}s+1)K_{\mathrm{p}}{K}_{\mathrm{t}}}=\frac{1}{J{s}^2+(B+K_{\mathrm{p}}{K}_{\mathrm{t}}{\tau }_{\mathrm{d}})s+K_{\mathrm{p}}{K}_{\mathrm{t}}}$$

(2)

${\theta }_{\mathrm{m}}$, $T_{\mathrm{l}}$, B, J, $K_{\mathrm{p}}$, ${\tau }_{\mathrm{d}}$ and $K_{\mathrm{t}}$ are the mechanical angle, load torque, motor damping constant, moment of inertia of the rotor, active impedance controller integrator time constant and motor torque constant, respectively. Comparing the coefficients between Equations (1) and (2) reveals the analogies depicted in

Table 1. Coefficients of the impedance loop, Mass-Spring-Damper and standard 2nd order transfer functions (Equation (1) and (2)).

Impedance Loop	Mass-Spring-Damper	Standard Form
J	$M_{\mathrm{s}}$	${\omega }_{\mathrm{n}}^{-2}$
$B+K_{\mathrm{p}}{K}_{\mathrm{t}}{\tau }_{\mathrm{d}}$	$B_{\mathrm{s}}$	$2\zeta {\omega }^{-1}$
$K_{\mathrm{p}}{K}_{\mathrm{t}}$	$K_{\mathrm{s}}$	1

.

This makes sense intuitively since moment of inertia J is essentially angular mass and the total system damping must be determined by the damping applied by the active impedance controller $K_{\mathrm{p}}{K}_{\mathrm{t}}{\tau }_{\mathrm{d}}$ and the damping constant of the motor (assuming an isolated system with no load). However, the transfer function of the PD controller which is included in Equation (2) is depicted in Equation (3).

$$\begin{array}{ccc}\hfill G_{\mathrm{PD}}=K_{\mathrm{p}}({\tau }_{\mathrm{d}}s+1)& \hfill & \hfill G_{\mathrm{lead}}=\frac{K_{\mathrm{p}}({\tau }_{\mathrm{d}}s+1)}{\alpha {\tau }_{\mathrm{d}}s+1},\alpha <1\end{array}$$

(3)

As the complex frequency goes towards infinity ($s\to \infty$), the magnitude of the PD controller goes towards infinity. Hence, high frequency motor angle noise gets amplified and injected into the torque loop. Thus, the lead compensator depicted in (3) is proposed, by which a pole at $\frac{1}{\alpha ·{\tau }_{\mathrm{d}}}\left[\frac{\mathrm{rad}}{\mathrm{s}}\right]$ is placed to reduce the gain at high frequencies. This should be placed about four to 10 decades higher than the maximum possible zero $\frac{1}{{\tau }_{\mathrm{d}}}$ to ensure a precise damping constant as well as a suitable reduction of high frequency magnitudes.

4. Novel Observer: Mechanical Angle, Electrical Angle and Speed Filtering

As depicted in Appendix J, noise arising from the angle sensor (typically a 12-bit magnetic encoder) and the online computation of speed is injected nearly directly into the motor phases when high-bandwidth torque control is enabled. Thus, an approach to filter the mechanical angle, electrical angle and speed is needed. A novel closed-loop observer, including sensor feedback, is proposed to ensure precise estimation with minimum delay, see Figure 5. Similar to a Luenberger observer, it consists of two steps—the estimate-ahead step and the correct step.

4.1. Estimation Step

As shown in the model, Equation (A8) (in Appendix B), the quadrature voltage $v_{\mathrm{q}}$ includes a term that includes the magnitude of the back-EMF (${\omega }_{\mathrm{m}}{\lambda }_{\mathrm{m}}{i}_{\mathrm{q}}$) which is proportional to the speed ${\omega }_{\mathrm{m}}$. Therefore, the observer estimates the speed ahead ${\widehat{\omega }}_{\mathrm{m}}^{-}$, by using the quadrature voltage reference $v_{\mathrm{q}}^{*}$, computed by the PI controller. Doing so requires the observer to subtract the voltage generated based on the internal line resistance and inductance $v_{\mathrm{RL}}$ from the quadrature voltage reference $v_{\mathrm{q}}^{*}$. The direct way to do this is to propagate the quadrature current $i_{\mathrm{q}}$ through the inverse of the electrical model ($L_{\mathrm{s}}s+R_{\mathrm{s}}$). However, as the inverse electrical model includes a zero it will amplify the noise coming from the quadrature current $i_{\mathrm{q}}$. Hence, a noise-reduced quadrature current estimation is computed instead by filtering the quadrature current reference $i_{\mathrm{q}}^{*}$ using a filter with the same bandwidth as the closed-loop torque controller. After subtracting the estimated RL voltage ${\widehat{v}}_{\mathrm{RL}}$ from the quadrature current reference, the inverse of the peak magnetic flux ${\lambda }_{\mathrm{m}}^{-1}$ is multiplied with the estimated back-EMF $\widehat{e}$, thereby predicting the mechanical speed.

The results shown in Figure 6 suggest that the observer provides good open-loop estimation (graph in lower right corner) of speed with enhanced high-frequency noise response in relation to that obtained by taking the derivative of the angle (graph in upper right corner), directly measured by the sensor. Note that the spikes of the speed estimated ahead ${\widehat{\omega }}_{\mathrm{m}}^{-}$ occur due to the big step in torque reference (by which the observer becomes very sensitive to model uncertainties). This will never occur in compliant robotics where the robot is operated in angle mode using the impedance loop instead of torque mode.

4.2. Correction Step

Assuming that the speed and angle obtained from the sensor provides the best low-frequency response (closest to the true angle and speed) and the worst high-frequency response (high-noise contribution), a correction step should be applied that ensures the observer tracks the low-frequency response of the sensor (the true angle) while including the low-noise (high-frequency response) of the estimation (${\widehat{\omega }}_{\mathrm{m}}^{-}$). This should be enabled by the correction loop in the observer. Applying the superposition principle to the correction loop in Figure 5 and including the a priori estimate of the angle ${\theta }_{\mathrm{m}}^{-}$ into the output with respect to estimation loop, the two loops can be derived (Figure 7).

Applying Mason’s formula to both loops, the closed-loop response in the laplace-domain of each loop is approximated:

$$\frac{{\widehat{\theta }}_{\mathrm{m}}\left(s\right)}{{\widehat{\theta }}_{\mathrm{m}}^{-}\left(s\right)}=\frac{s}{s-l}\frac{{\widehat{\theta }}_{\mathrm{m}}\left(s\right)}{{\theta }_{\mathrm{m},\mathrm{n}}\left(s\right)}=\frac{l}{s-l}$$

It is evident that the correction step outputs a combination of the high-frequency response of the estimation and the low-frequency response of the sensor which is required in order to achieve noise-reduced estimation with low time delay (see graph in the middle lower graph in Figure 6).

This correction is essentially a first order Finite Impulse Response (FIR)-filter which determines the corrected estimated speed ${\widehat{\omega }}_{\mathrm{m}}$ by weighing the initial guess (the estimated angle ${\widehat{\omega }}_{\mathrm{m}}^{-}$) against the measured angle ${\widehat{\theta }}_{\mathrm{m}},\mathrm{n}$. The estimated mechanical angle ${\widehat{\theta }}_{\mathrm{m}}$ is calculated by taking the integral to the corrected speed ${\widehat{\omega }}_{\mathrm{m}}$ and the electrical angle ${\widehat{\omega }}_{\mathrm{e}}$ is estimated by multiplying the estimated mechanical ${\widehat{\theta }}_{\mathrm{m}}$ angle with the motor pole pairs P (graph in the left lower graph in Figure 6).

The observer gain l may be designed by placing the pole using the observer error dynamic equation:

$$\begin{array}{c}\hfill e(k+1)=-le\left(k\right)\end{array}$$

Using this observer, high-accuracy, low-noise is obtained when low-speed is required. However, high-speed may lead to a non-linear relationship between the quadrature voltage $v_{\mathrm{q}}$ and speed (especially if over-modulation occurs) which could lead to observer diversion, thus, controller instability. A solution to this could either include increasing the observer gain l when speed increases so that the observer trusts the measurements more at high-speed. Otherwise, the observer gain l may be changed into a PI controller which includes enhanced low-frequency tracking response.

5. Kalman/Luenberger Observer: Quadrature Current Filtering

As depicted in Appendix J, the noise appearing in the quadrature current is mainly determined by the current sensors. Hence, the novel observer presented earlier has only a small influence on the quadrature current noise. Therefore, another observer is proposed which aims to reduce this without considerably reducing the bandwidth of the torque and field controllers. The proposed method is based on the two standard and well-known observer methods, the Luenberger observer and the Kalman filter. The gain may either be deterministically (Luenberger [29]) or stochastically (Kalman [30]) derived. The observer is shown in Figure 8.

The observer calculates the reference voltage that is applied across the internal line resistance and inductance $v_{\mathrm{RL}}^{*}$ by subtracting the back-EMF coupling term (shown in Figure 3) from the quadrature voltage reference $v_{\mathrm{q}}^{*}$. This is used to estimate the quadrature current $i_{\mathrm{q}}$ ahead by using the first order discretized state-space model of the voltage/current relationship in the internal RL circuit of the motor:

$$\dot{i_{\mathrm{q}}}\left(t\right)=\stackrel{A}{\overbrace{\phantom{{\displaystyle \frac{A}{A}}}-\frac{R_{\mathrm{s}}}{L_{\mathrm{s}}}}}{i}_{\mathrm{q}}\left(t\right)+\stackrel{B}{\overbrace{\phantom{{\displaystyle \frac{A}{A}}}\frac{1}{L_{\mathrm{s}}}}}{v}_{\mathrm{RL}}\left(t\right)$$

(4)

$R_{\mathrm{s}}$, $L_{\mathrm{s}}$, A and B are the internal line-resistance, internal line-inductance, state-transition gain and input gain, respectively. When a new measurement arrives, the quadrature current estimation is corrected by weighing the predicted estimate (apriori estimate ${\widehat{i}}_{\mathrm{q}}(k+1)$) against the measured quadrature current $i_{\mathrm{q},\mathrm{n}}$. A high Kalman gain $K_{\mathrm{k}}$ results in increased trust in the measurement with decreased trust in the estimate and vice-versa. $Q_{\mathrm{k}}$ and $R_{\mathrm{k}}$ is the variance of the noise arising in the quadrature voltage estimation and measured quadrature current, respectively. They can both be determined by measuring the noise in the system at steady-state (preferably at 0 $\mathrm{A}$). The Kalman gain is determined such that the Mean Squared Error ($P\left(k\right)$) goes towards zero, hence the noise is optimally filtered. However, instead of the stochastic derivation of the Kalman gain, one may instead choose to determine a fixed Luenberger gain $L_{\mathrm{k}}$ which can be done in a number of ways [31,32]. One way to do this is by placing the pole of the observer, using the observer error dynamic equation:

$$e(k+1)=(A_{\mathrm{k}}-L_{\mathrm{k}}{C}_{\mathrm{k}})e\left(k\right)$$

(5)

e is the observation error. The output gain $C_{\mathrm{k}}$, the discrete state transition gain $A_{\mathrm{k}}$ and the input $B_{\mathrm{k}}$ can be estimated using Equation (A9) (in Appendix B).

6. Experimental Test Setups

During all experiments, the motor module including test stand, T-motor U10PLUS KV80 and controller PCB (illustrated in Figure A5) are used and a DC voltage supply (TENMA 72-2640) is used to supply the motor controller PCB with 25 $\mathrm{V}$. All static parameters used for the experiments are shown in

Table A1. Specifications of the U10PLUS KV80 motor and the controllers.

Description	Reference	Unit	Value
Internal Line Resistance	R	$\mathrm{m}$$\mathsf{\Omega }$	95
Internal Line Inductance	L	$\mathsf{\mu }$$\mathrm{H}$	$63.7$
Max Continuous Current @180 $\mathrm{s}$	$I_{\max }$	$\mathrm{A}$	33
Max Continuous Power @180 $\mathrm{s}$	$P_{\max }$	$\mathrm{k}$$\mathrm{W}$	$1.5$
Nominal Excitation Voltage	$V_{\mathrm{DC}}$	$\mathrm{V}$	40
Peak Stall Torque	$T_{\max }$	$\mathrm{N}$$\mathrm{m}$	$3.6$
Max Operating Temperature	$T_{\mathrm{C},\max }$	${}^{\circ }\mathrm{C}$	180
Dimensions	DxT	$\mathrm{m}$$\mathrm{m}$	Ø89 × 40
Shaft Diameter	$D_{\mathrm{shaft}}$	$\mathrm{m}$$\mathrm{m}$	15
Weight	M	$\mathrm{g}$	500
Torque Constant	$K_{\mathrm{t}}$	$\frac{\mathrm{Nm}}{\mathrm{A}}$	$0.12$
Velocity Constant	$K_{\mathrm{V}}$	$\frac{\mathrm{RPM}}{\mathrm{v}}$	80
Number of poles	P	-	40
Moment of inertia on rotor	I	$\mathrm{k}$$\mathrm{g}$$\mathrm{m}$2	$0.00021$
Motor viscous damping	B	$\frac{\mathrm{N}\mathrm{m}}{\mathrm{rad}/\mathrm{s}}$	$0.000348$
Torque controller proportional gain	$K_{\mathrm{p}}$	$\frac{\mathrm{V}}{\mathrm{Nm}}$	$0.5806$
Torque controller integrator gain	$K_{\mathrm{i}}$	$\frac{\mathrm{V}}{\mathrm{Nms}}$	$819.5635$
Speed controller proportional gain	$K_{\mathrm{p},\omega }$	$\frac{\mathrm{N}\mathrm{m}}{\mathrm{rad}/\mathrm{s}}$	$0.545$
Speed/Angle observer gain	l		1500
Luenberger observer gain	$L_{\mathrm{k}}$		$0.4$
Sampling/PWM period	$T\mathrm{s}$	$\mathsf{\mu }$$\mathrm{s}$	40
Magnetic encoder resolution	-	bits	12

. The real hardware as well as all experimental setups can be observed in Figure 9. Appendix L includes a description of the electronics and software developed and implemented for the motor controller.

7. Experimental Results and Discussion

7.1. Experimental Results: Active Impedance Controller Compared with Dynamic Impedance Model

For these tests, the 1-link loadexperimental setup shown in Figure 9a, is used. The motor is driven in active impedance control mode with a fixed angle reference. Then, the shaft is forced out of the desired angle by rotating the propeller $\pi$ $\mathrm{rad}$(using an external force). The propeller is then released, allowing the controller to rotate the propeller back to the desired angle. This experiment is performed eight times using the eight different impedance characteristics listed in

Table A2. Impedance model and controller parameters.

	Impedance Model Parameters		Impedance Controller Parameters
#	Spring Constant	Damping Constant	Proportional Gain	Derivative Gain	Attenuation Factor
	${\mathit{K}}_{\mathbf{s}}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}}$]	${\mathit{B}}_{\mathbf{s}}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}/\mathbf{s}}$]	${\mathit{K}}_{\mathbf{p},\mathit{\Omega }}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}}$]	${\mathit{K}}_{\mathbf{d},\mathit{\Omega }}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}/\mathbf{s}}$]	α
Test 1	$0.1$	$0.0029$	$0.8382$	$0.0255$	$0.0125$
Test 2	1	$0.0029$	$8.3822$	$0.00255$	$0.1247$
Test 3	2	$0.0029$	$16.7645$	$0.0013$	$0.2495$
Test 4	3	$0.0029$	$25.1467$	$0.00085$	$0.3742$
Test 5	2	$0.0029$	$16.7645$	$0.0013$	$0.2495$
Test 6	2	$0.0097$	$16.7645$	$0.0047$	$0.0681$
Test 7	2	$0.0193$	$16.7645$	$0.0095$	$0.0336$
Test 8	2	$0.0290$	$16.7645$	$0.0143$	$0.0222$

. The results from the experiments are shown in Figure 10a,b.

As can be seen, the experimental results roughly match the mathematical models. Small deviations occur, both regarding the natural frequency and damping factor. As the natural frequency in a standard 2nd order under-damped rotational system mostly depends on the moment of inertia and the spring constant, the estimated inertia may be a bit off. The assumption of an infinite torque bandwidth may have an impact on the spring constant as well. The discrepancy of the damping may be caused by an inaccurate inertia estimation or by un-modeled damping in the system. These results highlight the importance of accurate estimation of the physical parameters (inertia, damping factor, etc.) and that the accuracy of the active impedance controller design depends on how close the impedance bandwidth is to the lead compensator and closed-loop torque controller poles.

Note, two videos showing the conducted experiments and their results are available as supplementary materials—with (Video S10) and without observers (Video S9).

7.2. Experimental Results: Torque Control—Angle and Speed Observer vs. no Observer

This experiment is conducted to show the results from deploying the novel angle and speed observer described in Section 4. The experiment is conducted under zero load as this results in high torque specific acceleration. This demands a lot from the observer, especially in terms of stability and accuracy. The motor controller is configured to run with torque control and a multiple torque reference step profile is applied. The results are shown in Figure 11. The torque rise time of about 133 $\mathsf{\mu }$$\mathrm{s}$, depicted in Figure 11a,b shows that high-bandwidth (approximately $2.6$ $\mathrm{k}$$\mathrm{Hz}$) is accomplished while relatively low overshoot occurs, verifying high robustness. This bandwidth is not as high as the torque controller bandwidth in the MIT Cheetah Mini ($4.5$ $\mathrm{k}$$\mathrm{Hz}$) [12]. The main reason for this is that the MIT Cheetah Mini is using the STM32F446 microcontroller, which enables twice the clock frequency than the TMS320F28069 (used in this project). Therefore, MIT are capable of running the controller loop at a sample rate of 40 $\mathrm{k}$$\mathrm{Hz}$ whereas 25 $\mathrm{k}$$\mathrm{Hz}$ is used in this work. However, the controller design proposed in this paper can deliver approximately the same bandwidth in relation to sample rate as enabled in the MIT Cheetah Mini. It must be emphasized that this bandwidth can be adjusted by adjusting the phase margin, where 60${}^{\circ }$ is used in this case to enable highly robust control. A large noise reduction can be observed in the speed ${\omega }_{\mathrm{m}}$, direct current $i_{\mathrm{d}}$ and phase voltages. As quantified by the noise analysis (Appendix J), using the noise-reduced estimated electrical angle greatly reduces the noise appearing in the direct-current which then reduces the noise on the phase voltages. It should be emphasized that the effect of the noise reduction is even more noticeable when using active impedance control and/or back-EMF feedforward. The reason for this is that the noise is then amplified (based on the speed calculation) and afterwards injected into the system (based on the high-bandwidth torque control).

Further investigation of the angle and mechanical observer performance is shown in Figure 12 both with (Inertia load) in Figure 9b) and without load.

As shown in Figure 12a,b, the main noise contribution arising in the 12-bit encoder is quantization noise. Therefore, the observer includes the effect of providing a higher resolution angle (both with and without load) than is provided by the sensor alone. One may argue that similar results may be obtained by picking a higher resolution magnetic encoder. However, today’s off-the-shelf high-end magnetic encoders are typically 12–14 bit, where the 14-bits encoders include noise on the two least significant bits meaning that the effective resolution is reduced to about 12 bits. The high frequency noise gets amplified when speed is calculated which is clearly depicted in Figure 12c. The observer is again capable of delivering a highly reduced noise response (both with and without load). A standard low-pass filter with the same amount of filtering would cause a huge phase-delay which the observer clearly does not. Therefore, it can be concluded that the proposed novel observer is capable of greatly reducing noise in the system while maintaining a high-bandwidth and highly robust closed-loop torque control.

To illustrate the behavior of the novel speed and angle observer, the intermediate results as well as the final output of the observer are shown in the observer diagram, see Figure 6. This behavior is further described in Section 4.

7.3. Experimental Results: Torque Control—Torque Observer vs. no Observers.

This experiment is conducted to investigate the performance of the torque observer proposed in Section 5. The motor controller is configured for torque control and multiple torque reference steps are performed. The results are shown in Figure 13.

As shown, high-frequency noise appearing on the quadrature current $i_{\mathrm{q},\mathrm{n}}$ (directly derived from noisy measurements) is significantly reduced by the observer ${\widehat{i}}_{\mathrm{q}}$. However, small discrepancies (offset errors) occur which are determined by model inaccuracies. However, the effect of this quadrature current discrepancy should not be significant when using active impedance control as the outer loop compensates for this issue.

7.4. Experimental Results: Speed Control—Angle, Speed and Torque Observer vs. no Observers

This experiment is conducted to investigate the performance of the entire proposed observer strategy including both the torque and speed/angle observers. Again, the test is conducted with zero load. The results are shown in Figure 14.

The effect of both observers on noise is consistently noticeable on all collected data. Figure 14g,h quantify a significant reduction in noise of approximately $13.5$ dB which corresponds with almost five times reduction. However, a small deviation occurs in the dynamic response of speed ${\omega }_{\mathrm{m}}$ and the quadrature current $i_{\mathrm{q}}$ which could be a result of small inaccuracies in the torque observer model.

The author of [12] noticed audible noise in the motor module at $4.5$ $\mathrm{k}$$\mathrm{Hz}$ and therefore decided to perform experiments at 1 $\mathrm{k}$$\mathrm{Hz}$ instead. By using the observers, proposed in this paper, this noise will be greatly reduced and the bandwidth of $4.5$ $\mathrm{k}$$\mathrm{Hz}$ can therefore be retained.

Note that two videos showing the conducted experiments and their results are available as supplementary materials (the flywheel in the videos is not connected in this experiment)—with (Video S10) and without observers (Video S9). Additionally, the tests have been conducted with variable speeds (Video S5 and S6) and with load 3 $\mathrm{N}$$\mathrm{m}$ (Video S3 and S4).

7.5. Experimental Results: Active Impedance Control with Angle/Speed and Torque Observers—Impact Force Load

This experiment is conducted to showcase the results from using the novel impact force load as well as the performance of the motor controller during impacts. The impact force load shown in Figure 9c is used, where a 3 $\mathrm{k}$$\mathrm{g}$ iron mass is connected to the linear rail and released at a distance from the impact block of 30 $\mathrm{c}$$\mathrm{m}$. The impedance model and controller parameters are listed in

Table A3. Impedance model and controller parameters.

	Impedance Model Parameters		Impedance Controller Parameters
#	Spring Constant	Damping Constant	Proportional Gain	Derivative Gain
	${\mathit{K}}_{\mathbf{s}}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}}$]	${\mathit{B}}_{\mathbf{s}}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}/\mathbf{s}}$]	${\mathit{K}}_{\mathbf{p},\mathit{\Omega }}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}}$]	${\mathit{K}}_{\mathbf{d},\mathit{\Omega }}$ [$\frac{\mathbf{N}\mathbf{m}}{\mathbf{rad}/\mathbf{s}}$]
Test 1	$0.1$	$0.029$	$0.0029$	$0.0029$

. The resulting torque, speed and angle are depicted in Figure 15.

Again, relatively high correlation between the mathematical model and the experimental results can be observed. The controller is capable of achieving impact force mitigation similar to that provided by the model. In addition, the controller runs stable and smoothly throughout the impact which sums up the controller and observer designs as high-performance. However, oscillations occur—this is one of the by-products of structural compliance resulting from the mechanical couplings depicted in Figure A3. Aside from the oscillations, structural compliance reduces torque transparency which results in lowering the force bandwidth at the end-effector. This is an issue in high-bandwidth requiring robotics. Hence, it must be emphasized that the structural compliance should be kept as low as possible for such applications. One way to reduce this issue is to include the robot kinematics and dynamics in the active impedance controller loop and employing an adaptable Kalman filter to estimate the Jacobian matrix online, as proposed and verified in [33]. Otherwise, the structural compliance can be reduced by using mechanical reinforcements or by choosing stiff materials.

Note that the two videos showing the conducted experiments and their results are available as supplementary materials—with stiff (Video S7) and soft active impedance control (Video S8).

7.6. Experimental Results: Active Impedance Control with Angle/Speed and Torque Observers—Compliance Test

Two experiments are conducted to show how the active impedance controller, including both observers, performs during a human-robot collision. For these tests, the 1-link load shown in Figure 9a is used. A sinusoidal angle reference is applied to the active impedance controller. Then, after a few cycles, a human hand is positioned between the start and end angle in the trajectory path for about three cycles, which ensures three collisions. This test was conducted two times—once with reduced torque controller bandwidth (50 $\mathrm{Hz}$) and once with the originally designed torque controller ($2.6$ $\mathrm{k}$$\mathrm{Hz}$). The reduced bandwidth of 50 $\mathrm{Hz}$ is arbitrarily chosen to show the importance of high torque bandwidth control when deployed as inner loop in an active impedance controller. [10] describes in detail the relationship between the torque controller bandwidth and the performance of active impedance control. The results are shown in Figure 16. It can be observed that when using the reduced torque bandwidth, active the impedance controller is unable to perform a precise reference tracking. Additionally, during impacts (gray areas), the active impedance controller is unable to mitigate the impact force, as high quadrature current spikes occur. Using the high-bandwidth torque controller designs, proposed in this article, the active impedance controller is capable of performing a precise reference tracking as well as absorbing the impact force. It must be emphasized, that in many cases, such as slow robot motion or low active impedance control bandwidth, 50 $\mathrm{Hz}$ may be adequate. However, in the case of high bandwidth interaction (e.g., cobot-human collision at fast motion or fast locomotion of legged robots), the high bandwidth is critical in order to ensure no damage occurs to the robot or the environment.

Note that two videos showing the conducted experiments and their results are available as supplementary materials—with low torque bandwidth (Video S11) and high torque bandwidth (Video S12).

8. Conclusions

In this work, a complete controller design scheme for high-bandwidth active impedance control of compliant robotics have been presented.

Design equations, which enables calculation of the parameters in the outer active impedance control loop (PD controller), is derived in Section 3.2. In Section 7.1, the design equations was shown to be precise for a variety of different spring constants and damping factors. However, the precision of the method is highly dependent on the precision of the used passive parameters (inertia, damping factor, etc.) and how close the impedance bandwidth is to the lead compensator and closed-loop torque controller poles.

In Section 3.1, design equations as well as best practice design of the field and torque loops are proposed. As verified in Section 7.2, the torque loop was capable of fast reference tracking (about $2.6$ $\mathrm{k}$$\mathrm{Hz}$ closed-loop bandwidth) while the field loop was capable of keeping the field at zero. This bandwidth is not as high as the torque controller bandwidth in the MIT Cheetah Mini ($4.5$ $\mathrm{k}$$\mathrm{Hz}$) [12]. The main reason for this is that the MIT Cheetah Mini is using the STM32F446 microcontroller which enables twice the clock frequency than the TMS320F28069 (used in this project). Therefore, it is capable of running the controller loop at a sample rate of 40 $\mathrm{k}$$\mathrm{Hz}$ whereas 25 $\mathrm{k}$$\mathrm{Hz}$ is used in this work. However, the controller design proposed in this paper enables approximately the same bandwidth in relation to sample rate as enabled in the MIT Cheetah Mini. It must be emphasized that this bandwidth can be adjusted by adjusting the phase margin, where 60${}^{\circ }$ is used in this case to enable highly robust control.

In Appendix J, the noise in a Field-Oriented Controlled BLDC motor was analyzed. This showed that the noise appearing on the quadrature current (in the torque loop) can be significantly reduced by reducing the noise on measured phase currents. In addition, the noise appearing in the field and impedance loop can be reduced by reducing noise on the measured rotor angle.

To address this, a Kalman/Luenberger observer was proposed to reduce noise in the torque loop and a novel speed and angle observer was proposed to reduce noise in the impedance and field loops. Section 7.2 showed that the novel angle and speed observer was capable of greatly reducing noise arising in the angle sensor (or increasing the resolution) at rotor speed at least up to 60 $\mathrm{rad}$/$\mathrm{s}$ while no noticeable delay occurred (as it would with a low-pass filter). This was shown to be the case both at zero load and at $0.000279$ $\mathrm{k}$$\mathrm{g}$m². Section 7.3 showed a great noise reduction on the quadrature current is achieved when deploying the torque observer. Section 7.4 showed that by using both observers $13.5$ dB (about five times reduction) noise reduction is facilitated, which concludes that the combination of high-bandwidth low-noise torque control of a BLDC motor has been achieved. The author of [12] noticed audible noise in the motor module at $4.5$ $\mathrm{k}$$\mathrm{Hz}$ and therefore decided to perform experiments at 1 $\mathrm{k}$$\mathrm{Hz}$ instead. Using the observers, proposed in this paper, this noise will be greatly reduced and the bandwidth of $4.5$ $\mathrm{k}$$\mathrm{Hz}$ can therefore be retained.

Section 7.5 and Section 7.6 showed that the entire controller scheme, including all three loop designs and both observers, was capable of performing well during demanding impact force tests. Additionally, Section 7.6 includes both a test using a reduced torque controller bandwidth (50 $\mathrm{Hz}$) and the original bandwidth. The results highlight the importance of high closed-loop torque bandwidth, as the reduced bandwidth was not capable of tracking the angle reference or absorbing the impact force properly.