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

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Compliant Actuators

#### 1.2. Existing Impedance Control Systems

#### 1.3. Work Contribution

- 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.

## 2. Proposed Active Impedance Controller System

## 3. Motor Controller Designs

#### 3.1. Torque and Field Controller Design

#### 3.2. Active Impedance Controller Design

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

#### 4.1. Estimation Step

#### 4.2. Correction Step

## 5. Kalman/Luenberger Observer: Quadrature Current Filtering

## 6. Experimental Test Setups

## 7. Experimental Results and Discussion

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

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

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

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

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

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

## 8. Conclusions

^{2}. 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.

## Supplementary Materials

- Video S1: 30 $\mathrm{rad}$/$\mathrm{s}$ speed step test without observers (0 $\mathrm{N}$$\mathrm{m}$ load)
- Video S2: 30 $\mathrm{rad}$/$\mathrm{s}$ speed step test with observers (0 $\mathrm{N}$$\mathrm{m}$ load)
- Video S3: 30 $\mathrm{rad}$/$\mathrm{s}$ speed step test without observers (3 $\mathrm{N}$$\mathrm{m}$ load)
- Video S4: 30 $\mathrm{rad}$/$\mathrm{s}$ speed step test with observers (3 $\mathrm{N}$$\mathrm{m}$ load)
- Video S5: Multi speed step test without observers 0 $\mathrm{N}$$\mathrm{m}$ load)
- Video S6: Multi speed step test with observers 0 $\mathrm{N}$$\mathrm{m}$ load)
- Video S7: Impact force test (stiff control)
- Video S8: Impact force test (soft control)
- Video S9: Active impedance control test without observers
- Video S10: Active impedance control test with observers
- Video S11: Human-robot collision test low bandwidth
- Video S12: Human-robot collision test high bandwidth
- C code S13: Entire motor controller source code
- PCB files S14: Motor Module 10S

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Clarke and Park Transformation

## Appendix B. BLDC Motor Model in the Rotating Reference Frame

## Appendix C. Controller Pseudo Algorithm

## Appendix D. Specifications of the Test Configuration

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 $\mathrm{s}$ | ${I}_{\mathrm{max}}$ | $\mathrm{A}$ | 33 |

Max Continuous Power @180 $\mathrm{s}$ | ${P}_{\mathrm{max}}$ | $\mathrm{k}$$\mathrm{W}$ | $1.5$ |

Nominal Excitation Voltage | ${V}_{\mathrm{DC}}$ | $\mathrm{V}$ | 40 |

Peak Stall Torque | ${T}_{\mathrm{max}}$ | $\mathrm{N}$$\mathrm{m}$ | $3.6$ |

Max Operating Temperature | ${T}_{\mathrm{C},\mathrm{max}}$ | ${}^{\circ}\mathrm{C}$ | 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 |

## Appendix E. Impedance Test 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 | $0.0290$ | $16.7645$ | $0.0143$ | $0.0222$ |

## Appendix F. Impact Test 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 | $0.1$ | $0.029$ | $0.0029$ | $0.0029$ |

## Appendix G. Derivation of Torque and Field Controller Design Equations

## Appendix H. Symbolic Expressions of the Sensitivity Factors

#### Appendix H.1. Direct and Quadrature Current with Respect to Measured Angle

#### Appendix H.2. Direct and Quadrature Current with Respect to Measured Currents

## Appendix I. Field Oriented Control

**Figure A2.**Illustration of the FOC commutation strategy and resulting waveforms. The turquoise box outlines the Digital Signal Processor (DSP) tasks used in FOC.

## Appendix J. Noise Analysis Based on the Sensitivity Method

- High-end and off-the-shelf magnetic encoders are typically maximum 12-bit resolution
- The direct current is highly affected by angle sensor noise
- The Proportional-Derivative (PD) controller amplifies angle sensor noise which is almost directly injected into the motor phases (due to high-bandwidth torque control)
- If speed feed-forward is required (to decouple the torque loop from the back-EMF), the angle sensor value is once again amplified and injected directly into the motor phases

## Appendix K. Impact Force Benchmark Test Configuration

**Figure A3.**Illustration of the test configuration used for testing the motor controller during impact forces.

## Appendix L. Electronics and Software Platform

**Figure A4.**Illustration of all relevant communication between the TMS320F28069 DSP and the external components which are utilized in the motor controller PCB (manufactured at University of Southern Denmark (SDU)).

**Figure A5.**Rendered 3D models of the entire motor module hardware, including (

**a**) the U10 PLUS KV80 motor and (

**b**) the motor controller PCB developed at SDU. The motor module is mounted on the 3D print model test stand.

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**Figure 1.**Illustration of five compliant actuators. The figures are replicated from the three corresponding figures in [1]. The sample delay listed below refers to the delay, between an occurring impact to an opposing force that has been applied, introduced using a digital impedance controller.

- (a)
**Geared Motor with Force/Torque sensor (GMS)**embeds a low-diameter motor and a high gear ratio transmission to actuate the robot link. A torque or force sensor is used as feedback to perform torque or force control. Pros:variable impedance response, high torque density. Cons: sample delay (cannot mitigate high speed shocks), low open-loop force controller bandwidth (at end-effector), low Z-width [10], low force transparency.- (b)
**Series Elastic Actuator (SEA)**includes a low-diameter motor and a high gear ratio transmission to actuate the robot link. A spring is connected between the gears and the end-effector of the robot link. Pros: high torque density, high energy efficiency, simple control, no sample delay* (can mitigate high speed shocks). Cons: fixed impedance response [9], it offers improved force transparency and open-loop force controller bandwidth (at end-effector) in relation to GMS and PEA [11], but is not comparable to the proprioceptive actuator design on this matter [1,9,12,13].- (c)
**Proprioceptive actuator**includes a high-diameter motor and a low gear ratio transmission to actuate the robot link. As the transparency between the motor shaft and the end-effector is very low (due to low gearing and high stiffness), the motor phase currents can be used to estimate the torque at the end-effector (proprioception). Pros: high open-loop force controller bandwidth (at end-effector), high force transparency [1,9,12,13], high Z-width [10]. Cons: complex control, sample delay* (cannot mitigate high speed shocks), lower energy efficiency and torque density than other electromagnetic actuators [1].- (d)
**Parallel Elastic Actuator (PEA)**includes a low-diameter motor and a high gear ratio transmission to actuate the robot link. Additionally, it includes a spring which is connected in parallel with the robot link. Pros: high torque density, high efficiency [14], simple control, no sample delay* (can mitigate high speed shocks). Cons: fixed impedance response, low open-loop force controller bandwidth (at end-effector), low force transparency.- (e)
**Variable Stiffness Actuator (VSA)**is typically comprised of the same elements as SEA, where an additional motor is controlling the stiffness of the elastic element. Pros: high torque density, high energy efficiency [14], variable impedance response, no sample delay* (can mitigate high speed shocks). Cons: complex control, low open-loop force controller bandwidth (at end-effector), low force transparency.

**Figure 2.**Simplified block diagram of the entire closed loop motor controller system, including Field Oriented Control, angle and current observers, virtual compliance and torque and field controller loops.

**Figure 3.**Simplified block diagram illustrating the rotating reference frame equivalents of the Field, Torque and Impedance loop.

**Figure 4.**Illustration of the hip compliance (${K}_{\mathrm{s},\mathrm{hip}}$, ${B}_{\mathrm{s},\mathrm{hip}}$) and foot compliance (${K}_{\mathrm{s},\mathrm{leg}}$, ${B}_{\mathrm{s},\mathrm{leg}}$) in a robot leg (

**a**) and of the simplified impedance loop in relation to external force $\frac{{\theta}_{\mathrm{m}}}{-{T}_{\mathrm{l}}}$ (

**b**). (

**a**) is adopted from [10] and (

**b**) is derived and simplified from Figure 3

**Figure 5.**Illustration of the torque loop and speed and angle observer proposed in this section. The light-yellow boxes outline the algorithms used for the observer. The decoupling term is proposed to be 0 $\mathrm{V}$ but can be included if needed.

**Figure 6.**Illustration of the speed and angle observer including test results of the internal parameters. The first axis of all plots relates to time. The reference speed ${\omega}_{\mathrm{m}}^{*}$ is an estimate of the true speed, which is calculated by applying a 200 $\mathrm{Hz}$ low-pass filter to the derivative of the raw angle ${\dot{\theta}}_{\mathrm{m},\mathrm{n}}$ and then offset in time with an amount corresponding with the time-delay introduced by the filter.

**Figure 7.**Simplified block diagram of the output vs. measurement loop $\frac{{\widehat{\theta}}_{\mathrm{m}}\left(k\right)}{{\theta}_{\mathrm{m},\mathrm{n}}\left(k\right)}$ (

**a**) and the output vs. estimation loop $\frac{{\widehat{\theta}}_{\mathrm{m}}\left(k\right)}{{\widehat{\theta}}_{\mathrm{m},\mathrm{n}}^{-}\left(k\right)}$ (

**b**).

**Figure 8.**Illustration of the torque loop and torque observer proposed in this section. The light-yellow boxes outline the algorithms used for the observer. The decoupling term is proposed to be 0 $\mathrm{V}$ but can be included if needed.

- (a)
**1-link load**includes a 30 $\mathrm{c}$$\mathrm{m}$ carbon fiber tube (propeller) attached to the motor shaft through a hollow cylindric plastic spacer. The purpose of this configuration is to demonstrate active impedance by applying external force to the propeller- (b)
**Inertia load**includes a 900 $\mathrm{g}$ cylindric iron load directly attached to the motor shaft. The purpose of this setup is to demonstrate the proposed controller including a fixed level of attached inertia.- (c)
**Impact load**includes a weight which is mounted on a linear rail. The motor is connected to another rail through a torque sensor (T22/20NM), two mechanical couplings and a spur gear. The purpose of this setup is to demonstrate how the motor controller complies during impact force.

**Figure 10.**Measured (dashed lines) and simulated (solid lines) results from eight experiments (listed in Table A2) conducted on the active impedance controller. The spring constant is varied in (

**a**) and the damping constant is varied in (

**b**). The measured results are offset and scaled to match the amplitude and start angle of the simulated.

**Figure 11.**Measured and estimated results from two conducted experiments using torque mode with multiple step in torque reference with zero load—without observer (

**a**,

**c**,

**e**) and with the novel angle and speed observer (

**b**,

**d**,

**f**). The torque rise times are depicted on (

**a**,

**b**). Both speed estimations ${\omega}_{\mathrm{m},\mathrm{n}}$ and ${\widehat{\omega}}_{\mathrm{m}}$ are filtered with the same low-pass filter including a 3 dB bandwidth of 200 $\mathrm{Hz}$ for better performance visualization.

**Figure 12.**Measured and estimated results from the conducted experiment using torque mode with multiple step in torque reference with zero load (

**a**,

**c**) and with $0.000279$ $\mathrm{k}$$\mathrm{g}$ $\mathrm{m}$

^{2}(

**b**,

**c**)—including the novel angle and speed observer. The zoomed areas depict the noise reduction and dynamic capabilities of the observer. Both speed estimations ${\omega}_{\mathrm{m},\mathrm{n}}$ and ${\widehat{\omega}}_{\mathrm{m}}$ are filtered with the same low-pass filter including a 3 dB bandwidth of 200 $\mathrm{Hz}$ for better performance visualization.

**Figure 13.**Measured and estimated results from the conducted experiment using torque mode with multiple step in torque reference with zero load—including the novel angle and speed and the Kalman/Luenberger observer. The zoomed areas depict the noise reduction capabilities of the Kalman/Luenberger observer.

**Figure 14.**Measured and estimated results from two conducted experiments using speed mode with a single step in speed reference with zero load—without observer (

**a**,

**c**,

**e**,

**g**) and with both the novel angle and speed and the Kalman/Luenberger observers (

**b**,

**d**,

**f**,

**h**). Both speed estimations ${\omega}_{\mathrm{m},\mathrm{n}}$ and ${\widehat{\omega}}_{\mathrm{m}}$ are filtered with the same low-pass filter including a 3 dB bandwidth of 200 $\mathrm{Hz}$ for better performance visualization.

**Figure 15.**Measured (${T}_{\mathrm{meas}}$, ${\omega}_{\mathrm{m},\mathrm{meas}}$ and ${\theta}_{\mathrm{m},\mathrm{meas}}$) and simulated (${T}_{\mathrm{simu}}$, ${\omega}_{\mathrm{m},\mathrm{simu}}$ and ${\theta}_{\mathrm{m},\mathrm{simu}}$) results from the experiment (listed in Table A3) conducted on the active impedance controller. (

**a**) ${T}_{\mathrm{meas}}$, (

**b**) ${\omega}_{\mathrm{m},\mathrm{meas}}$, (

**c**) ${\theta}_{\mathrm{m},\mathrm{meas}}$. The load configuration, proposed in Appendix K, is used with 3 $\mathrm{k}$$\mathrm{g}$ mass released from a distance of 10 $\mathrm{c}$$\mathrm{m}$. The simulated results are offset and scaled to match the amplitude and start angle of the measurements. The torque ${T}_{\mathrm{meas}}$ is estimated from the quadrature current by ${T}_{\mathrm{meas}}={K}_{\mathrm{t}}{i}_{\mathrm{q}}$.

**Figure 16.**Measured and estimated results from two conducted experiments using active impedance mode with a sinusoidal angle reference and with a 30 $\mathrm{c}$$\mathrm{m}$ propeller. (

**a**,

**c**) is at reduced torque controller bandwidth (50 $\mathrm{Hz}$) and (

**b**,

**c**) is at the originally designed torque controller (50 $\mathrm{Hz}$). Both tests are performed including both the novel angle and speed observer and the torque observer. The three gray areas on each plot depict three human-robot collisions. These are performed by putting a human hand in the middle of the trajectory path for about three trajectory cycles.

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 |

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**MDPI and ACS Style**

Lund, S.H.J.; Billeschou, P.; Larsen, L.B. High-Bandwidth Active Impedance Control of the Proprioceptive Actuator Design in Dynamic Compliant Robotics. *Actuators* **2019**, *8*, 71.
https://doi.org/10.3390/act8040071

**AMA Style**

Lund SHJ, Billeschou P, Larsen LB. High-Bandwidth Active Impedance Control of the Proprioceptive Actuator Design in Dynamic Compliant Robotics. *Actuators*. 2019; 8(4):71.
https://doi.org/10.3390/act8040071

**Chicago/Turabian Style**

Lund, Simon Hjorth Jessing, Peter Billeschou, and Leon Bonde Larsen. 2019. "High-Bandwidth Active Impedance Control of the Proprioceptive Actuator Design in Dynamic Compliant Robotics" *Actuators* 8, no. 4: 71.
https://doi.org/10.3390/act8040071