Performance improvement of a six-degree-of-freedom elastic pendulum thrust stand for micropropulsion

Abstract

This study presents the design and performance evaluation of an updated six-degree-of-freedom elastic pendulum thrust stand with enhanced measurement capabilities. The thrust stand is designed to measure all thrust and torque components, which are critical for advanced missions utilizing miniature propulsion systems. Key enhancements include the integration of high-resolution electrostatic capacitive displacement sensors and the expansion of measurable test targets. The thrust stand employs a null-balance method and was successfully demonstrated with a few mN-class cold-gas thruster. The results showed significant improvements in measurement resolution, with uncertainties of thrust vector components within 0.13 mN and those of torque components within 0.028 mN$\cdot$m for a 4 mN thrust level. This research aims to establish a practical and versatile six-degree-of-freedom measurement system applicable to a wide range of propulsion systems, including electric propulsion systems.

Introduction

In recent years, research and development of miniature propulsion systems for CubeSats have expanded exponentially to support diverse and sophisticated missions [1,2,3]. Propulsion systems are essential for orbital maneuvers and angular momentum management. However, unintended thrust and torque components, caused by side thrust, assembly errors, or misalignments, can significantly impact spacecraft operations. Electric propulsion systems, such as ion thrusters and Hall thrusters, introduce additional challenges including swirl torque, whose cumulative effects during extended operations can be substantial [4,5,6,7]. Measuring all thrust components in 6 DoF (Degrees of Freedom)-three thrust and three torque components-is thus essential for precise control, especially for miniature systems constrained by limited redundancy, weight, and space. A few studies have addressed 6-DoF thrust measurement for CubeSat-sized propulsion systems. Kakami et al. developed a magnetically levitated thrust stand designed to measure 1 N-class propulsion systems weighing approximately one kilogram, making it suitable for large-thrust systems [8]. However, measuring lower-thrust systems with this thrust stand is challenging due to its inherent limitation in measurable weight range. This limitation arises from the low thrust-to-weight ratio of miniature propulsion systems, making it difficult to accurately measure thrust components in all directions while physically supporting the propulsion system.

Moriai et al. proposed a 6-DoF elastic pendulum thrust stand that uses springs to support the propulsion system. This approach increases the effective thrust-to-weight ratio, enabling more accurate measurements in lower thrust ranges and heavier systems up to 5 kg. The thrust stand employs a null-balance method, where thrust and torque are determined based on the driving forces of the actuators, while displacements and angular displacements are controlled to zero. For a 10 mN-class propulsion system, the reported measurement uncertainties were 0.82 mN for thrust and 0.13 mN$\cdot$m for torque, respectively [9, 10]. Despite these advancements, accurately measuring electric propulsion systems, which typically operate in the sub-mN range, remains challenging. One of the dominant limitations is the resolution of the implemented LED displacement sensors (5 $\upmu$m resolution) used in the control. Additional challenges of the thrust stand include: 1) the size of the test target is restricted to less than 240 mm in length, 2) the test target’s weight is constrained by the pendulum structure weight, and 3) the alignment of the pendulum is highly technical and time-consuming due to its high degree of freedom.

In this paper, we present the design and testing of an updated elastic pendulum thrust stand with enhanced measurement performance. The primary improvement in enhancing measurement capability is the resolution of the displacement sensor, which improved to 96 nm by using electrostatic capacitive sensors. Additional three improvements are implemented for the pendulum structures: 1) accommodating larger targets by expanding the spacecraft mounting plate, 2) reducing the weight of the pendulum by removing the magnetic damper, and 3) streamlining the alignment process of the thrust stand by introducing an alignment unit. The performance of the thrust stand was demonstrated using a few mN-class cold-gas thruster. We report on the key challenges and areas for further enhancement by comparing measurement performance before and after the improvements. The ultimate goal of this research is to realize the practical application of the elastic pendulum thrust stand across a wide range of propulsion systems, including electric propulsion systems.

Updated 6-DoF elastic pendulum thrust stand

Mechanical design

Figure 1 shows the CAD image of the updated elastic pendulum thrust stand with a CubeSat model mounted on it. The thrust stand comprises an elastic pendulum, which consists of an aluminum mounting plate suspended by three springs and the test target (CubeSat/propulsion system), two sets of six displacement sensors for pendulum alignment and accurate measurement, and six actuators (voice coil motors, VCMs). The springs are coil springs made of SWP-A (MISUMI Model LWSHB-10–310). They were selected to support test targets with a mass of up to 5 kg. The inertial coordinate system, with its origin O and x, y, and z axes, is defined as shown in Fig. 1. Angular displacements $\phi$, $\theta$, and $\psi$ are defined about the x, y, and z axes, respectively.

Fig. 1

CAD image of the updated 6-DoF elastic pendulum thrust stand. The origin is located at the geometric center of the triangular mounting plate’s top surface, with the x, y, and z axes defined accordingly. The thrust stand includes six displacement sensor units and six actuators (voice coil motors, VCMs). The length of each side of the mounting plate is 420 mm

In the design of this thrust stand, four updates were made compared to the previous study [10]. First, the resolution of the displacement measurement system was significantly improved by adding electrostatic capacitive displacement sensors (96 nm resolution) in addition to the existing LED sensors (5 $\upmu$m resolution). Second, the length of one side of the spacecraft mounting plate was increased from 360 mm to 420 mm, allowing for the smooth accommodation of the test target up to 300 mm in length. Third, the magnetic damper of the elastic pendulum was eliminated, which reduced the weight of the pendulum while maintaining the maximum allowable weight for a test target, even with the expanded spacecraft mounting plate. Fourth, all displacement sensor units and VCMs were mounted on a single plate, which was attached to three-axis translational and rotational stages fixed inside a vacuum chamber. This configuration enables standardized alignment and improves usability.

Displacement sensor units

Each displacement sensor unit comprises an electrostatic capacitive sensor (UNIPULSE Model PS-III-6N) for null-balance control and an LED sensor (OMRON Model Z4D-F04A) for the rough alignment of the elastic pendulum. Since LED sensors have a measurement range five times wider than that of electrostatic capacitive sensors, they were used for the alignment of the pendulum, ensuring the proper positioning of the VCMs and displacement sensors. The displacement $\varvec{x}=(x, y, z)^\mathrm{T}$ and the angular displacement $\varvec{\phi }=(\phi , \theta , \psi )^\mathrm{T}$ of the pendulum can be expressed as:

$$\begin{aligned} \left( \begin{array}{c} \varvec{x} \\ \varvec{\phi } \end{array}\right) =\varvec{A}\varvec{d}, \end{aligned}$$

(1)

where $\varvec{A}$ is a constant 6$\times$6 matrix, and $\varvec{d}$ represents the displacement sensor signals from the electrostatic capacitive sensors (6$\times$1 vector) [10]. The electrostatic capacitive sensors have a peak-to-peak resolution of 96 nm.

VCM (Voice Coil Motor)

Figure 2 shows a cross-sectional view of a VCM, a non-contact electromagnetic actuator. Each VCM generates a static magnetic field, enabling it to produce a force proportional to the coil current within the magnetic field. The force-to-current ratio of the VCMs varies from 0.16 to 0.19 N/A, calibrated using an electronic balance as described in [10]. The magnetic field generated by each VCM is designed to be contained within the VCM, ensuring that the external magnetic field remains sufficiently small (<1 mT) and has a negligible impact on test targets. Figure 3 illustrates the positions and positive directions of the VCMs within the thrust stand. VCM1-3 are aligned along the $+z$ axis, while VCM4-6 are positioned on the xy plane. Together, the VCMs can generate any desired thrust vector or torque components acting on the pendulum. According to [10], the combined force vector produced by the VCMs, $\varvec{F}_\mathrm{VCM}$ (3$\times$1 vector along x, y and z), is given by:

$$\begin{aligned} \varvec{F}_\mathrm{VCM} = \sum \limits _{k=1}^{6} \varvec{a}_k \alpha _k i_k, \end{aligned}$$

(2)

where $\varvec{a}_k$ is the 3$\times$1 unit direction vector of VCM-k ($|\varvec{a}_k|=1$), $\alpha _k$ is the force-to-current ratio of VCM-k, and $i_k$ is the coil current of VCM-k. Similarly, the combined torque vector produced by the VCMs, $\varvec{T}_\mathrm{VCM}$ (3$\times$1 vector about x, y and z), is expressed as:

$$\begin{aligned} \varvec{T}_\mathrm{VCM} = \sum \limits _{k=1}^{6} \left( \varvec{r}_k \times \varvec{a}_k \right) \alpha _k i_k, \end{aligned}$$

(3)

where $\varvec{r}_k$ is the position vector of VCM-k relative to the center of mass of the pendulum. The coil current $i_k$ can be described as:

$$\begin{aligned} i_k = u_k V_k + w_k, \end{aligned}$$

(4)

where $V_k$ is the voltage applied to VCM-k, and $u_k$ and $w_k$ are constants.

Fig. 2

A cross-sectional view of a VCM (Voice Coil Motor). The yoke part and the coil part are not in contact

Fig. 3

Positions and orientations of the VCMs within the thrust stand. Yellow arrows indicate the positive direction of each VCM

Dynamics and control system

The thrust stand utilizes the null-balance method for the measurement. In this method, the actuators are controlled to “null” the displacements of the pendulum, ensuring that the total driving force and torque generated by the actuators are equal to the thrust force and torque produced by the test target. Since the null-balance method assumes that position, velocity, and acceleration are zero, the equation of motion can be expressed as:

$$\begin{aligned} \left( \begin{array}{c} \varvec{F} \\ \varvec{T} \end{array}\right) = - \left( \begin{array}{c} \varvec{F}_\mathrm{VCM} \\ \varvec{T}_\mathrm{VCM} \end{array}\right) + \Delta \varvec{\varepsilon }, \end{aligned}$$

(5)

where $\Delta \varvec{\varepsilon }$ is the error during null-balance control (6$\times$1 vector). The thrust stand employs a PID controller for null-balance control, using feedback from the VCM voltages $\varvec{V}=(V_1, V_2, ..., V_6)^\mathrm{T}$. The control input $\varvec{V}$ is determined as follows:

$$\begin{aligned} \varvec{V}= \varvec{K}_\mathrm{c}\left( \begin{array}{c} \Delta \varvec{d} \\ \int \Delta \varvec{d} \textrm{d}t \\ \frac{\textrm{d}}{\textrm{d}t}\Delta \varvec{d}\end{array}\right) , \end{aligned}$$

(6)

where $\varvec{K}_\mathrm{c}$ is the 6$\times$18 feedback gain matrix determined using the pole placement method, and $\Delta \varvec{d}$ represent deviations from their respective target values [10]. Since this PID controller had nonzero D-gain parameters, the VCMs could partially function as virtual magnetic dampers. Considering the steady state during thruster operation, $\Delta \varvec{\varepsilon }$ can be expressed as:

$$\begin{aligned} \Delta \varvec{\varepsilon } = -\varvec{K} \varvec{A} \Delta \varvec{d} = -\varvec{B}\Delta \varvec{d}, \end{aligned}$$

(7)

where $\varvec{K}$ is the 6$\times$6 stiffness matrix of the pendulum, $\varvec{B}$ is the 6$\times$6 constant matrix, and $\Delta \varvec{\varepsilon }$ and $\Delta \varvec{d}$ represent time-averaged values. The matrix $\varvec{K}$ was obtained experimentally by applying known thrust and torque to the pendulum, as described in detail in [10].

Experimental setup

The thrust stand was demonstrated by measuring a few mN-class cold-gas thruster. Here, a brief overview of the thruster is provided, while comprehensive detail can be found elsewhere [10]. Figure 4(a) shows the system diagram of the thruster, and Fig. 4(b) displays an image of the thruster mounted on the thrust stand, with its nozzle oriented along the z-axis. The thruster comprises a CO$_2$-gas propellant tank, a pressure regulator, a pressure sensor, a valve, and a nozzle. The nozzle has a throat diameter of 0.4 mm, an expansion ratio of 4, a divergence angle of 30$^\circ$, and a convergence angle of 90$^\circ$. The nozzle inlet pressure was set to 25 kPa. Additionally, the thruster is equipped with batteries and a microcomputer for wireless thruster monitoring and control. This allows the thruster to operate completely wirelessly, eliminating potential interference from harness connections. The center of mass of the thruster was experimentally determined with an uncertainty of ±3 mm, as described in [10]. According to nozzle theory, a cold-gas thruster generates the thrust $|\varvec{F}|$ as

$$\begin{aligned} |\varvec{F}| = A_\mathrm{t} p_\mathrm{nozzle} C_F, \end{aligned}$$

(8)

where $A_\mathrm{t}$ is the nozzle throat area, $p_\mathrm{nozzle}$ is the nozzle inlet pressure, and $C_F$ is the thrust coefficient [11].

Fig. 4

(a) System diagram of the cold-gas thruster used in the experiment [10]. Solid lines represent gas piping. (b) Image of the thruster mounted on the thrust stand’s mounting plate

The experiment was conducted in a steel vacuum chamber with dimensions of 1.2 m in diameter and 1.7 m in length. The chamber was evacuated using a rotary pump (ULVAC Model VD401), and the background pressure was kept below 35 Pa during thruster operation. The background pressure in this experiment was determined to ensure choked flow at the nozzle throat of the cold-gas thruster. However, the thrust stand can, in principle, operate in a much higher vacuum ($<1\times 10^{-2}$ Pa). The pressure was monitored using a vacuum gauge (Pfeiffer Vacuum Model PKR251).

Experimental and analytical method

Cold-gas thruster measurement

The cold-gas thruster described in Section 3 was tested to demonstrate and evaluate the performance of the thrust stand. The thruster was mounted on the thrust stand inside the vacuum chamber, with its nozzle oriented along the $+z$ axis. It was operated for approximately 30 seconds, during which its thrust vector and torque were measured using the null-balance control. The control and measurement were performed at a sampling rate of 100 Hz using LabVIEW.

During thruster operation, the center of mass of the pendulum was assumed to change due to propellant consumption. The position vector of VCM-k relative to the center of mass of the pendulum, $r_k(t)$, was adjusted based on the mass flow rate, which was assumed to be proportional to the nozzle inlet pressure.

The measured thrust and torque included the effects of this gradual mass change. To isolate the thrust and torque generated purely by the propulsion system, $\varvec{F}_\mathrm {meas.}$ and $\varvec{T}_\mathrm {meas.}$, the time-dependent effects of the mass change were calculated and subtracted from the raw measurements $(\varvec{F}_\mathrm{raw}, \varvec{T}_\mathrm{raw})^\mathrm{T} = -(\varvec{F}_\mathrm{VCM}, \varvec{T}_\mathrm{VCM})^\mathrm{T}$. This time-dependent thrust and torque isolation was performed in all directions, ensuring that the derived thrust and torque values accurately represent those generated by the propulsion system alone.

Uncertainty analysis

An uncertainty analysis was conducted to evaluate the measurement performance and compare it to the previous thrust stand. From Eq. 5, the j-th component of thrust generated by the CubeSat/propulsion system, $F_{(j)}$ ($j=1,2,3$, corresponding to $F_x$, $F_y$ and $F_z$), is expressed as:

$$\begin{aligned} F_{(j)} = -F_{\textrm{VCM}, j} + \Delta \varepsilon _j, \end{aligned}$$

(9)

and the j-th component of torque, $T_{(j)}$ ($j=1,2,3$, corresponding to $T_x$, $T_y$ and $T_z$), is similarly expressed as:

$$\begin{aligned} T_{(j)} = -T_{\textrm{VCM}, j} + \Delta \varepsilon _{j+3}, \end{aligned}$$

(10)

where ${F_{\textrm{VCM},j}}$, ${T_{\textrm{VCM},j}}$ and $\Delta \varepsilon _j$ represent the j-th components of $\varvec{F}_\mathrm{VCM}$, $\varvec{T}_\mathrm{VCM}$, and $\Delta \varvec{\varepsilon }$, respectively. Hence, the uncertainties of $F_{(j)}$ and $T_{(j)}$, $\sigma _{F_{(j)}}$ and $\sigma _{T_{(j)}}$, are given by:

$$\begin{aligned} \sigma _{F_{(j)}} = \sqrt{\sigma _{F_{\textrm{VCM},j}}^2 + \sigma _{\Delta \varepsilon _j}^2}, \end{aligned}$$

(11)

and

$$\begin{aligned} \sigma _{T_{(j)}} = \sqrt{\sigma _{T_{\textrm{VCM},j}}^2 + \sigma _{\Delta \varepsilon _{j+3}}^2}, \end{aligned}$$

(12)

respectively, where $\sigma _{F_{\textrm{VCM},j}}$ and $\sigma _{T_{\textrm{VCM},j}}$ represent the uncertainties in the thrust and torque generated by the VCMs, and $\sigma _{\Delta \varepsilon _j}$ represents the uncertainty in null-balance control. These uncertainties were calculated to the first order as:

$$\begin{aligned} \sigma _{f_j} = \sqrt{\sum \limits _{k=1}^6 \left[ \sum \limits _{l=1}^N \left( \frac{\partial f_j}{\partial q_{l}}\sigma _{q_l}\right) ^2\right] }, \end{aligned}$$

(13)

where $f_j$ represents a physical quantity that is a function of variables $q_l$. Here, $k=1$ to 6 corresponds to VCM-k for $\sigma _{F_\mathrm{VCM}}$ and $\sigma _{T_\mathrm{VCM}}$, and to displacement sensor unit numbers and the corresponding column of $\varvec{B}$ for $\sigma _{\Delta \varepsilon }$. Details of the uncertainty analysis can be found in [10], and the parameters used for the uncertainty analysis are listed in Table 1.

Table 1 Error budget for the uncertainty analysis, including the maximum uncertainties of each parameter and their descriptions. Uncertainty values $\sigma _\alpha$, $\sigma _r$, $\sigma _U$, $\sigma _V$, and $\sigma _w$ use the same values as [10]

Results and discussions

Cold-gas thruster measurement

Figure 5(a) shows the time history of displacements during the thruster measurement demonstration. The thruster began operating at $t=0$ s and stopped at $t=30$ s. While some transient responses in $\Delta d_1$, $\Delta d_2$, and $\Delta d_3$ were observed due to the pendulum’s reaction when the thruster was turned on or off, all displacements were generally controlled to within ±0.005 mm of zero. Figure 5(b) depicts the time history of the controlled coil current of the VCMs during the null-balance method. Since the nozzle was oriented along the z-axis, VCM1-3 exhibited significant responses, with VCM3 showing the largest reaction due to the nozzle’s closer proximity to VCM3. Additionally, offsets in some VCMs’ currents after the thruster operation were observed, attributed to the change in propellant mass. Figure 5(c) presents the converted raw thrust and torque values, $\varvec{F}_\mathrm{raw}$ and $\varvec{T}_\mathrm{raw}$, derived from the VCM coil currents. Gradual changes in $F_z$, $T_x$, and $T_y$ were observed due to the decrease in propellant mass, which totaled 0.27 g. Other components, such as $F_y$ and $T_x$, were also present during thruster operation.

Fig. 5

Time histories of (a) displacement sensor deviations from the target values, (b) coil current of the VCMs, and (c) raw thrust and torque measurements. In (b), the linear drifts in the coil currents of VCM1-3 were removed throughout the operation. A moving average with a window of 0.8 seconds was applied to the data in (b) and (c)

Figure 6 shows the pure force and torque generated by the thruster operation, $\varvec{F}_\mathrm {meas.}$ and $\varvec{T}_\mathrm {meas.}$, after removing the time-varying components caused by mass loss. The thrust vector is likely influenced by a combination of lateral thrust and alignment errors. Similarly, the generation of torques can be attributed to disturbance torques resulting from the misalignment between the thrust vector and the center of mass of the pendulum, as well as alignment errors, as discussed in [10].

Fig. 6

Time history of (a) nozzle inlet pressure of the cold-gas thruster, and measured values $\varvec{F}_\mathrm {meas.}$ and $\varvec{T}_\mathrm {meas.}$ in (b) $F_x$, (c) $F_y$, (d) $F_z$, (e) $T_x$, (f) $T_y$, and (g) $T_z$. (b) to (g) exclude the effect of mass change, providing purely generated thrust and torque by the thruster

Uncertainty analysis

Figure 7 presents the measured thrust and torque values as bar graphs, with the maximum uncertainties during the thruster operation, $\sigma _{F_{(j)}}$ and $\sigma _{T_{(j)}}$, represented as error bars. Each thrust and torque value is calculated as the time-averaged value within the range excluding the transient response (approximately $t=5$ to 30 s in Fig. 6). Each thrust and torque value, along with its uncertainty, is compared to the results from the previous thrust stand [10]. While the absolute measured values differ between this study and the previous experiment, the uncertainties have been significantly reduced in all thrust and torque components with the updated thrust stand. For example, the proportion of uncertainty in $F_z$ was $\sim 9$% in the previous thrust stand, whereas it has been reduced to $\sim 3$% in this study. Overall, we achieved uncertainties of less than 0.13 mN for thrust vector components and less than 0.028 mN$\cdot$m for torque components.

Fig. 7

Comparison of the measured thrust and torque between this study (yellow box graph) and the previous study (blue box graph) [10]. The error bars represent the maximum uncertainties in each direction during the thruster operation

The theoretical thrust value calculated using Eq. 8 was $5.0\pm 3.0$ mN, while the measured thrust value was $4.0\pm 0.13$ mN, showing reasonable agreement. The uncertainty in the theoretical thrust was calculated from the uncertainty in the pressure sensor, whereas the uncertainty in the measured value was determined from $\sigma _{F_{(j)}}$.

The proportion of the $\sigma _{\Delta \varepsilon _j}$ to the total uncertainty, $P_{\Delta \varepsilon _j}$, is defined as:

$$\begin{aligned} P_{\Delta \varepsilon _j} = \frac{\sigma _{\Delta \varepsilon _j}^2}{\sigma _{F_{(j)}}^2} \end{aligned}$$

(14)

and

$$\begin{aligned} P_{\Delta \varepsilon _{j+3}} = \frac{\sigma _{\Delta \varepsilon _{j+3}}^2}{\sigma _{T_{(j)}}^2}, \end{aligned}$$

(15)

for the thrust vector and torque components, respectively ($j=1,2,3$). Similarly, the proportion of $\sigma _{F_{\textrm{VCM}},j}$ and $\sigma _{T_{\textrm{VCM}},j}$ to the total uncertainty, $P_{F_{\textrm{VCM},j}}$ and $P_{T_{\textrm{VCM},j}}$, are given by:

$$\begin{aligned} P_{F_{\textrm{VCM},j}} = \frac{\sigma _{F_{\textrm{VCM},j}}^2}{\sigma _{F_{(j)}}^2}, \end{aligned}$$

(16)

and

$$\begin{aligned} P_{T_{\textrm{VCM},j}} = \frac{\sigma _{T_{\textrm{VCM},j}}^2}{\sigma _{T_{(j)}}^2}, \end{aligned}$$

(17)

respectively ($j=1,2,3$). Figure 8 compares the uncertainty values in all directions between (a) this study and (b) the previous study [10]. In the updated thrust stand, more than 97% of the uncertainties in all thrust and torque components are attributed to $P_{F_\mathrm{VCM}}$ or $P_{T_\mathrm{VCM}}$, compared to 10–60% in the previous thrust stand. This result indicates that the uncertainty arising from errors in the null-balance control has been significantly reduced by improving the displacement sensor resolutions. However, there remains room for further improvement. For $P_{F_\mathrm{VCM}}$, depending on the direction, the dominant contributions to uncertainty are $\sigma _\alpha \gg \sigma _V, \sigma _U, \sigma _w$. Similarly, for $P_{T_\mathrm{VCM}}$, the contributions follow $\sigma _r>\sigma _\alpha \gg \sigma _V, \sigma _U, \sigma _w$. This suggests that reducing $\sigma _\alpha$ and $\sigma _r$ could further enhance measurement accuracy. The primary cause of $\sigma _\alpha$ is the relative position misalignment between the VCM coil and yoke as discussed in [10]. Ensuring precise relative positioning using additional sensing devices or redesigning VCMs to minimize relative mobility could improve measurement performance. Similarly, $\sigma _{r}$ could be reduced by ensuring the absolute position of the VCM coil and yoke relative to the center of mass of the pendulum.

Fig. 8

Comparison of the thrust and torque uncertainty between (a) this study and (b) previous study [10]. The proportion of the uncertainty in null-balance control $P_{\Delta \varepsilon }$, and that of the uncertainty in the force and torque generated by the VCMs, $P_{F_\mathrm{VCM}}$ and $P_{T_\mathrm{VCM}}$, are shown as purple bars and light blue bars, respectively

Applicability for electric propulsion systems

Although the proposed thrust stand has proven effective for measuring a few-mN cold-gas thruster, some challenges must be considered when measuring electric propulsion systems. The first challenge is the power supply for the thruster. Due to the high degree of freedom of this thrust stand, the measurement assumes that the propulsion system is modular and self-contained. Consequently, the operation of thrusters that require a continuous and significant power supply is limited when using batteries. To address this issue, we have proposed a wireless power transfer system that enables semi-permanent thruster operation without affecting thrust measurements [12]. This system utilizes RF-based magnetic field resonance to wirelessly charge batteries. Another potential challenge is the heat generated by the electric propulsion system. If the heat emitted from the thruster alters the mechanical properties of the pendulum during null-balance control, it may lead to the detection of false thrust or torque. Possible countermeasures include thermal insulation in the pendulum’s thermal design and radiative cooling to mitigate heat generation. These considerations, in addition to the uncertainty improvement strategies discussed in the previous section, are essential for accurately measuring electric propulsion systems in 6 DoF.

Conclusion

In this paper, an updated version of a six-degree-of-freedom elastic pendulum thrust stand was designed and tested. Several modifications were made to the previous design, including enhancing the resolution of the displacement sensors, enlarging the spacecraft mounting plate, eliminating the magnetic dampers, and adding alignment units. The thrust stand utilized the null-balance method with the new displacement sensors and successfully measured the thrust and torque components of a few mN-class cold-gas thruster. The measurement results revealed the presence of side thrust and disturbance torque components, indicating possible misalignments within the thruster. The measured thrust magnitude was consistent with the theoretical value. An uncertainty analysis of the measurements was performed and compared with the previous elastic pendulum thrust stand. For a propulsion system generating approximately 4 mN of thrust, the uncertainties were within 0.13 mN for thrust vector components and 0.028 mN$\cdot$m for torque components. The proportion of these uncertainties relative to the measured values was significantly reduced for all thrust and torque components, demonstrating the effectiveness of enhancing the displacement sensor resolution. Potential approaches to further reduce measurement uncertainties were discussed, such as ensuring the absolute and relative position accuracy of the voice coil motor components.