Quasi-Wireless Capacitive Power Transfer for Wire-Free Robotic Joints ^†

Abstract

Robotics is a highly active, multidisciplinary research area with a broad list of applications. A large research focus is to enhance modularity in order to expand kinematic capabilities, lower fabrication time, and reduce construction costs. Traditional wiring within a robot presents major challenges with mobility and long-term maintenance. Designing robotics without wires would make a significant functional impact. This work presents a new application of quasi-wireless capacitive power transfer that investigates impedance matching parameters over a highly resonant, coupled transmission line to achieve efficient power transfer over a robotic chassis. A prototype is developed and its operating metrics are analyzed with regard to the matching parameters.

1. Introduction

A robust number of wires is the typical solution when constructing a robot. Wiring is problematic and requires maintenance, as a single break can result in system failure, or worse a hazardous environment. Wireless Power Transfer (WPT) could be a possible means to remove wiring within robotic design [1,2,3,4,5,6,7,8]. Researchers have investigated Inductive Power Transfer (IPT), demonstrating a 1 degree of freedom experimentally [9], which was theoretically expanded to 3 degrees of freedom [10]. A major challenge with IPT comes from lower misalignment tolerances between the transmitter (Tx) and receiver (Rx). For power transfer within a robotic appendage, developing methods to maintain IPT alignment can be difficult and lead to limitations in motion. Additionally, there is still a bulk of wires and electronics (such as an inverter) that must be present at each linkage if IPT is used [3,11,12]. A second method of WPT that exists is Capacitive Power Transfer (CPT), where an electric field at high frequency is used to transfer power from one electrode “plate” to another. CPT approaches have two primary configuration methodologies: a bipolar method, where the Tx has two physical plates with a matching two plate Rx [13], and a unipolar (or single-wire) method, which consists of a single Tx plate and a ground plane with the Rx either sharing the Tx ground plane or having a capacitive connection to that ground plane through a counterpoise (a large return plate) [14]. For design of a fully wireless robotic system, the same issues arise that were present in IPT, a need for a bulk of wiring and an inverter between each wireless joint [15,16,17]. Due to space limitation on most robotic systems, it is generally not possible to include a large counterpoise for single-wire CPT. Therefore, it can be concluded that there is no existing WPT methodology that can remove the wiring completely within a robotic system.

Table 1. Advantages and disadvantages of various WPT configurations.

	IPT	MRPT	CPT	APT
Pros	Most commonly used WPT technique [18] Capable of high power transmission level (kW) [19] High power transfer efficiency [19,20] Low sensitivity to environmental factors (pollutants and weather) [21]	Employs resonance frequency matching of Tx and Rx, which allows for less reliance on alignment [18,19] Range can be extended using intermediate coils that are tuned to the same system’s resonant frequency [22,23]	Transfer power through metallic materials without significant eddy current losses [19,20,22] Less reliance on Tx and Rx alignment [18] Implementation is often more cost efficient than with IPT systems [20]	Most secure WPT methodology due to the safe transmission of sound waves in the required frequency band [21] Resistivity to electromagnetic interference due to the transfer of energy occurring through sound waves [18]
Cons	Potential of significant eddy current losses [19] Potential for cross-talk due to inductance leakage [23] Challenging to transmit power through objects such as walls [21]	Complex systems, potentially consisting of additional matching networks [18] High sensitivity to frequency shifts due to resonance frequency aligned Tx and Rx [18]	Bipolar CPT requires many capacitive plates for transmission [18] Single-Wire CPT requires a large counterpoise [14] Normally, lower transmission efficiency and distance than IPT [18,20]	Relatively low power transmission levels (mW), often being used in bio-medical devices [18] Hardly explored in robotics due to low transmission efficiency [18]

includes a side-by-side comparison of state-of-the-art WPT techniques, listing the pros and cons. It includes IPT, Magnetic Resonance Power Transfer (MRPT), CPT, and Acoustic Power Transfer (APT).

With this in mind, the work presented here aims to create a new method of robotic design that completely removes wiring along and between joints of a robotic chassis. A quasi-wireless capacitive (QWiC) method will be utilized that replaces the large counterpoise requirement of a single-wire CPT system with a small quarter wave resonator (QWR) acting as a Rx, providing a compact method of power transfer over the surface of a robotic chassis. This approach presents a significant deviation from the standard two-wire electrical design methodology, simplifying it to only one electrical connection per joint. In other words, the robot wiring will be completely replaced by a conductive chassis that will allow the QWR and chassis surface to act as a power conduit.

One challenge in utilizing the QWiC method is that the QWR is highly sensitive to external influences very near or touching the QWR receiver. This makes impedance matching significantly more difficult. Firstly, the impedance becomes position dependent, such that the parameters of the matching network are based on the location it is placed (similar to an antenna). Secondly, unlike an antenna and most problematic, when the matching network is connected, the internal coupling at each section of the QWR causes the entirety of the QWR’s transmission line characteristics to change, making a simple smith-chart impedance calculation nearly impossible. To address this issue, a less invasive technique will be presented that offers a way to improve the impedance match of the load, without drastically affecting the QWR’s transmission line characteristics.

2. Materials and Methods

Conventional IPT and CPT designs require a minimum of two impedance matched Tx and Rx elements with proper alignment to allow for efficient power transfer [24]. Typically, these systems are operated in a frequency range where the currents and voltages are uniform, allowing a lumped circuit model approach when making impedance matching calculations. However, a QWiC system operates in highly resonant mode where the distributed transmission line inductances at each section are mutually coupled to adjacent sections, forming what was defined as a mutually coupled transmission line ($\mu$cT-line) [25].

To reduce the impact of changes in the $\mu$cT-line, the approach will be to form “taps” along the QWR. The idea is to shunt more current into the load rather than the QWR using the tap. This form of shunt-based impedance matching attempts to create a parallel path that utilizes a capacitor to negate the effects of load on the QWR operation.

The voltage distribution along an untapped QWR’s $\mu$cT-Line, with respect to the circuit neutral, is found to be [25]

$$V=Z_T{I}_S(1-e^{-\varphi })$$

(1)

where $I_S$ is the current flowing through the QWR, $Z_T$ is the total impedance of the QWR that is exponentially distributed by $(1-e^{-\varphi })$, and $\varphi$ is the position along the QWR in radians. This expression is only valid when the QWR’s inductance is in resonance with the stray capacitance, which is the condition required to transfer power. The stray capacitance has the most impact at the top, or free end, of the QWR. The bottom of the QWR represents the majority of the distributed inductance. It is expected that Equation (1) will change with an applied tap.

To derive the impact a tap would make on the QWR, Figure 1 is used as the distributed model of the $\mu$cT-line with a matching network shunted at the lower end. The total current in the QWR can be split between an upper impedance ($Z_{UR}$) and a lower impedance ($Z_{LR}$). Segmenting the QWR into four discrete sections and applying loop analysis produces

$$[\frac{-j}{\omega x_1^{\prime }{C}_1}+j\omega x_1{L}_1]i_1+[\frac{j}{\omega x_1^{\prime }{C}_1}-\omega x_2{M}_{12}]i_2+[-j\omega x_1{L}_1]i_L=V_S,$$

(2)

$$[\frac{j}{\omega x_2^{\prime }{C}_2}+\omega x_2{M}_{21}]i_1+[\frac{-j}{\omega x_1^{\prime }{C}_1}+j\omega x_2{L}_2-\frac{j}{\omega x_2^{\prime }{C}_2}]i_2+[\frac{j}{\omega x_2{C}_2}-\omega x_3{M}_{23}]i_3=0,$$

(3)

$$[\frac{j}{\omega x_2^{\prime }{C}_2}+\omega x_2{M}_{32}]i_2+[\frac{-j}{\omega x_2^{\prime }{C}_2}+j\omega x_3{L}_3-\frac{j}{\omega x_3^{\prime }{C}_3}]i_3+[\frac{j}{\omega x_3^{\prime }{C}_3}-\omega x_4{M}_{34}]i_4=0,$$

(4)

$$[\frac{j}{\omega x_3^{\prime }{C}_3}+\omega x_3{M}_{43}]i_3+[\frac{-j}{\omega x_3^{\prime }{C}_3}+j\omega x_4{L}_4-\frac{j}{\omega x_4^{\prime }{C}_4}]i_4=0,$$

(5)

$$\begin{array}{c}\hfill [-j\omega x_1{L}_1]i_1+[-j\omega x_2{M}_{12}]i_2+[R_L^{\prime }+j\omega L_L-\frac{j}{\omega C_L}+j\omega x_1{L}_1]i_L=0,\end{array}$$

(6)

where $R^{\prime }$ is the equivalent resistance of the load that accounts for transformation and rectification. Equation (6) can be written in terms of $i_L$

$$\begin{array}{c}\hfill i_L=\frac{x[-j\omega x_1{L}_1]i_1+[-j\omega x_2{M}_{12}]i_2}{[R_L^{\prime }+j(\omega L_L-\frac{1}{\omega C_L}+j\omega x_1{L}_1)]}\end{array}$$

(7)

Dividing $i_L$ into its real and imaginary parts yields

$$i_L=\frac{j\omega L_1{x}_1{R}_L^{\prime }(I_1)+\omega L_1{x}_1(\omega L_L-\frac{1}{\omega C_L}\omega L_1{x}_1)(I_1)}{B}+\frac{j\omega x_2{M}_{12}{R}_L^{\prime }(I_2)+\omega x_2{M}_{12}(\omega L_L-\frac{1}{\omega C_L}\omega L_1{x}_1)(I_2)}{B}$$

(8)

where

$$\begin{array}{c}\hfill B={(R_L^{\prime })}^2+{(\omega L_L+\omega L_1{x}_1-\frac{1}{\omega C_L})}^2\end{array}$$

(9)

Substituting Equation (8) into (2), the real and imaginary parts become

Real:

$$\frac{{\omega }^2{x}_1^2{L}_1^2{R}_L^{\prime }(I_1)}{B}+\frac{{\omega }^2{L}_1{M}_{12}{x}_1{x}_2{R}_L^{\prime }(I_2)}{B}-V_S+\cdots$$

(10)

Imaginary:

$$\dots j(\omega L_1{x}_1-\frac{1}{\omega C_1{x}_1^{\prime }}-{\omega }^2{\gamma }_1)I_1+j(\frac{1}{\omega C_1{x}_1^{\prime }}+\omega M_{12}{x}_2-{\omega }^2{\gamma }_2)I_2=0,$$

(11)

where

$$\begin{array}{c}\hfill {\gamma }_1=\frac{L_1^2{x}_1^2(\omega L_L-\frac{1}{\omega C_L}+\omega L_1{x}_1)}{B};\end{array}$$

(12)

$$\begin{array}{c}\hfill {\gamma }_2=\frac{L_1{M}_{12}{x}_1{x}_2(\omega L_L-\frac{1}{\omega C_L}+\omega L_1{x}_1)}{B}.\end{array}$$

(13)

${\gamma }_1$ is the reactive effect of the tapped load branch current ($I_L$ in Figure 1) on the self inductance of the Lower Resonator, while ${\gamma }_2$ is the effect of $I_L$ on the mutual coupling between Section 1 and Section 2. At resonance, the reactive components of the $\mu$cT-line will cancel at each distributed section. Following the same procedure presented in [26] but for a tapped system, the reactive terms for each loop can be equated, where the capacitive elements can be used to link the self inductances to the mutual inductances in each equation, Equations (2)–(5). This represents a total resonant condition, which is experimentally observable in the QWR when at resonance. From Equation (11),

$$\begin{array}{c}\hfill \omega L_1{x}_1=-\omega M_{12}{x}_2+{\omega }^2{\gamma }_1+{\omega }^2{\gamma }_2.\end{array}$$

(14)

Solving for $M_{12}$,

$$\begin{array}{c}\hfill M_{12}=\frac{-L_1{x}_1+\omega {\gamma }_1+\omega {\gamma }_2}{x_2}.\end{array}$$

(15)

Substituting $M_{12}$ into the reactive components of (3),

$$\begin{array}{c}\hfill L_2{x}_2+M_{23}{x}_3=(\frac{-L_1{x}_1+\omega {\gamma }_1+\omega {\gamma }_2}{x_2})x_1.\end{array}$$

(16)

Solving for $M_{23}$,

$$\begin{array}{c}\hfill M_{23}=\frac{-L_2{x}_2^2-L_1{x}_1^2+\omega {\gamma }_1{x}_1+\omega {\gamma }_2{x}_1}{x_2{x}_3}.\end{array}$$

(17)

Substituting $M_{23}$ into the reactive parts of (4),

$$\begin{array}{c}\hfill L_3{x}_3+M_{34}{x}_4=M_{23}{x}_2=\frac{-L_2{x}_2^2-L_1{x}_1^2+\omega {\gamma }_1{X}_1+\omega {\gamma }_2{x}_1}{x_3}.\end{array}$$

(18)

Solving for $M_{34}$,

$$\begin{array}{c}\hfill M_{34}=\frac{-L_1{x}_1^2-L_2{x}_2^2-L_3{x}_3^2+\omega {\gamma }_1{x}_1+\omega {\gamma }_2{x}_1}{x_3{x}_4}.\end{array}$$

(19)

Finally, substituting $M_{34}$ into the reactive parts of (5),

$$\begin{array}{c}\hfill \omega L_4{x}_4-\omega (\frac{-L_1{x}_1^2-L_2{x}_2^2-L_3{x}_3^2+\omega {\gamma }_1{x}_1+\omega {\gamma }_2{x}_1}{x_4})=\frac{1}{\omega C_4{x}_4}.\end{array}$$

(20)

Rearranging

$${\omega }^2(L_1{x}_1^2+L_2{x}_2^2+L_3{x}_3^2+L_4{x}_4^2)-{\omega }^3{\gamma }_1{x}_1-{\omega }^3{\gamma }_2{x}_1=\frac{1}{C_4}$$

(21)

Equation (21) is similar to what is found in [26] but with additional $\gamma$ terms that subtract from the total additive inductance. Using the same procedure to derive Equation (21), the capacitive terms can be solved and equated to Equation (21). This yields an expression for the resonant condition of the QWR.

$$\begin{array}{c}\hfill \frac{(x_1+x_2)}{\omega C_1{x}_1^{\prime }}+\frac{(x_2+x_3)}{\omega C_2{x}_2^{\prime }}+\frac{(x_3+x_4)}{\omega C_3{x}_3^{\prime }}+\frac{1}{\omega C_4}=\omega (L_1{x}_1^2+L_2{x}_2^2+L_3{x}_3^2+L_4{x}_4^2)-{\omega }^2{x}_1{\gamma }_1\end{array}$$

(22)

Note that the ${\gamma }_2$ terms cancel in Equation (22). From Equation (22), it can be seen that at resonance, ${\gamma }_1$ acts to subtract the inductance of the QWR below the tap location ($x_1$ section). This causes the non-linear distributions of the $\mu$cT-line to start at the tap location and not at the connection of the supply. This is an important result, as it shows that the non-linear effects of the $\mu$cT-line have been greatly decoupled from the load branch, such that variations in the load and associated matching network will not greatly change the electrical length of the QWR once connected. When $X_C=X_L$, the most current will flow through the load branch, improving the efficiency of transmission.

The current of the QWR can be approximated by assuming that the non-linear voltage distribution occurs only in the upper portion of the QWR, above the tap, allowing the portion below the tap to be a paralleled equivalent of $Z_{LR}$ and $Z_L$.   

$$\begin{array}{c}\hfill I_{QWR}\approx \frac{V_S}{\frac{Z_{LR}+Z_L}{Z_L{Z}_{LR}}+Z_{UR}(1-e^{-\Phi })}\end{array}$$

(23)

It is important to validate the above theory. The current along the QWR is difficult to directly measure since probes placed very near the QWR will alter the stray capacitance, drastically changing the parameters of the $\mu$cT-line. Using thermal imaging, it is possible to approximate the distribution of current in the system. Referring to Figure 2, the heat in the image is from the $i^{2}R$ losses in the QWR; by plotting the thermal distribution from the image, the square root of this thermal distribution yields an approximate current along the QWR. The measured input current (I) was used to properly scale the data extracted from the thermal image. The predicted versus experimental measurements are shown in Figure 3, where the predicted current was plotted from Equation (23). The plots are in decent agreement but are not exact since the thermal distribution is not a direct, one-to-one measurement of current—only an approximation. Thermal conduction within the copper windings of the QWR will spread with time. This causes error when trying to approximate the value of current from the thermal image.

2.1. Experimental Setup

A 1-degree of freedom prototype was fabricated and is shown in Figure 4. The prototype consisted of a motor with an encoder, a QWR, a frame, and a concentric base plate. The frame was 3-D (Drawn using Autodesk Inventor Professoinal 2021 Build:421 and printed using Original Prusa i3 MK3S) printed using Polylactic acid (PLA) filament, which was then coated with conductive paint (841AR Super Shield Nickel Conductive Paint). Permanent magnets were attached to the bottom of the frame such that the the prototype would magnetically connect to the base plate. The base plate comprised an acrylic sheet that housed two electrodes: a steel “anode” plate and a grounded steel “cathode” plate. RF power was supplied to the center section of the base plate via a coaxial cable. Nylon M3 screws were used to secure the electrodes to the acrylic plate. Due to the high impedance operation of the QWR, a ferrite toroidal transform was used in the matching branch Figure 4a, while also providing isolation for the load. The matching transformer was composed of 20 gauge magnet wire and a Fair-Rite type 67 core. The toroid had an outer diameter of 35 mm, inner diameter of 22.5 mm, and a thickness of 13 mm Figure 5. The overall circuit is shown in Figure 4a, where the QWR was constructed of 176 turns of 22 gauge magnet wire with a total physical length of 135 mm and an outer diameter of 47.5 mm. Taps were placed every 15 turns along the entirety of the QWR.

2.2. Control Scheme

The prototype load consisted of the motor with integrated control electronics. Figure 6 shows the block diagram of the motor control system. The microcontroller was an ATmega 32u4. A DRV8833PWP dual H-bridge motor driver from Texas Instruments located in Dallas, TX, USA was used to drive a brushed DC N20 motor. Figure 6 is a photograph of the fabricated control PCB. The control board used an infrared (IR) NEC communication protocol to actuate the motor left or right. The NEC protocol is advantageous as it has readily available libraries for the ATmega microcontroller and offers a low bit error rate (BER) by averaging multiple signals sent for a command. The two TSOP382 IC’s receive the IR signals. A standard IR LED controller (such as found within a TV remote) was chosen as the transmitter.

A separate full bridge rectifier (Figure 7) in tandem with a 5V DC regulator was used to operate the embedded microcontroller. This was placed on a separate PCB that was positioned/stacked on top of the control board. Referring to Figure 7, the “V_HF+” and “V_HF-” are the inputs from the RF power signal that was connected to the tap/matching network of the QWR. The fabricated PCB is shown in Figure 7.

2.3. Unilateral Wiring Current Path

The current in the system follows a path that is along the chassis of the robot, over the motor housing/gearbox, and into the QWR. This current path will be defined as a unilateral wiring scheme, since there is no return wire anywhere in the system. Any thermal effects along the motor and associated gear box is an area of concern when being incorporated into the unilateral circuit. To investigate this, the skin depth was calculated to determine the penetration of the unilateral current flow. Using the standard equation for skin depth,

$$\delta =\sqrt{\frac{\rho }{\pi f\mu }},$$

(24)

where $\rho$ is the resistivity of the material the current flows over, f is the operating frequency, and $\mu$ is the permeability of material. Assuming the motor housing is a basic steel, $\mu$ can be estimated at approximately 1200${\mu }_0$ with a resistivity of $1\times {10}^{-7}$$\Omega$m. The skin depth is on the order of 3 $\mathsf{\mu }$m. This number is sufficient to warrant an experimental test with these motors to determine if any thermal heating occurs.

Figure 8 is a photograph of an experiment to test the heating through the motor. In this test, the motor was connected in series with an RF amplifier (The A150 RF Amplifier from Electronics and Innovation, LTD) and a QWR. This experiment was run for an hour, with 15 W being delivered to the QWR through the motor. The input current was measured at 474 ${\mathrm{mA}}_{RMS}$. The temperature of the coil was found to be 37.72 °C and the motor was 26.29 °C. This was compared to a room temperature of 23 °C. With only a 3 degree temperature rise over the course of the 1 h, the amount of thermal heating was insignificant. Therefore, it is possible to transfer power over the motor housing and subsequent gearbox with little thermal effects due to the QWR’s high-impedance operation. There are obvious limits to this approach as the system is scaled; it will become important to form capacitive links between the motor and adjacent joint. This is a focus for future work. However, the unilateral nature makes such capacitive linkages significantly easier than a bilateral CPT system. A bilateral CPT approach would require 2 transmitting plates and 2 receiving plates per linkage, increasing design complexity. The unilateral system needs only 1 transmitting plate and 1 receiving plate per linkage.

3. Results

Each tap location along the QWR was tested with the matching network, as shown in Figure 4a. The QWR was maintained at its resonant frequency for every tap location tested. These changes occur due to the reduction in the electrical length of the QWR. The total system efficiency ($\eta$) was calculated by measuring the DC power output at the rectifying PCB, and dividing it by the measured AC input power at the base plate.

$$\eta =\frac{P_{Load(DC)}}{P_{in(AC)}}=\frac{V_{DC}{I}_{DC}}{V_{RMS}{I}_{RMS}cos\theta }.$$

(25)

Figure 9 shows the voltage and current waveforms input to the base plate of the robotic system. From Figure 10, it can be seen that the percentage of the QWR that is tapped has a direct relationship to the efficiency. The presence of a capacitor in the load-side of the impedance branch also had a significant impact on the system transfer efficiency.

Without any capacitance to balance the load-side branch, the efficiency was the lowest of the experimental measurements, achieving a maximum transfer efficiency of 79%. Using a capacitor to balance load-side branch, the peak efficiency of 82% was achieved. As can be seen in the figure, for all the cases tested, the efficiency reached a point where it plateaued, while the efficiency increased by a marginal amount. This plateau occurs at a tap that spans roughly 25% of the coil’s length for the tests that had a matched impedance. The plateau for the test without a capacitor occurred at 43% of the coil’s length (nearly half the coil). Utilizing proper matching capacitance can reduce the amount of tap size required for power transfer on the QWR. This is significant since a higher efficiency can be achieved without shifting the frequency as much. This will allow for lower frequencies to be achieved with the same QWR.

Figure 11 plots the change in the resonant frequency with tap location. The data sets for the three capacitors used show that the capacitance does not have any large bearing on the shift in frequency as more of the coil is tapped. However, the number of turns of a tap produces a linear change in the operating frequency as the tap size is increased (Figure 11). This result shows that shunting a portion of the QWR using a tap creates a reduction in the electrical length in the $\mu$cT-line. The linear response in frequency vs. tap size is beneficial for designing a QWR to operate at a desired frequency while maintaining an improved power transfer efficiency.

The theory derived earlier in the manuscript assists in conveying what Figure 10 and Figure 11 show, which is that this topology can be used to improve the transfer efficiency while minimally affecting the operating frequency of the system. This can be a new framework for designing much more complex robots using impedance matching QWR topology to achieve better power transfer in the field of QWiC.

The ability to transfer power wirelessly over a robotic chassis, in a unilateral fashion, offers a unique design approach that eliminates wires between joints. Figure 12 depicts how such a unilateral wiring design can be expanded upon to operate any number of linkages using the QWiC power method. The multi-joint approach in Figure 12 is currently under development and will be communicated in a future work.

4. Discussion

A unique, unilateral current flow technique was demonstrated using a single degree of freedom to transfer power over the chassis of a robotic joint. This work explored the use of a matching capacitance in tandem with various tap locations along a QWR for improved efficiency using the single-wire QWiC method. From the efficiency data, it was determined that the tap location and matching capacitance allows one to achieve an improved power transfer scheme with a predictable effect on frequency shift of the QWR. The results from this study can be used to design a system that operates at both a specified frequency and a much higher transfer efficiency.

This novel quasi-wireless power technique opens a completely new design space, where wiring is eliminated within the chassis and between joints of machines, allowing for highly resilient systems that could have transformative impacts in robotics and their associated applications/industries.

Future work will explore different matching techniques beyond a single capacitor (such as LCC) in the load branch, along with capacitive methods of impedance transformation that do not require a transformer. Research is ongoing to improve various aspects of the design. Modulation of the RF power along the chassis could be used for communication, similar to [27].