Abstract
Recognition of the critical importance of positioning, navigation, and timing to all economic sectors is driving the development of diverse alternatives to global navigation satellite systems (GNSSs), termed AltPNT. One promising approach is to leverage the proliferation of small satellite constellations in low Earth orbit (LEO) to deliver GNSS augmentation services. The generation of one-way ranging signals suitable for AltPNT requires stable timing, accurately referenced to a common timescale such as Global Positioning System Time or Coordinated Universal Time. This paper describes a small-scale laboratory demonstration of an approach for cooperatively realizing a local timescale using low-size, weight, and power clocks distributed across multiple platforms, with no dependence on a GNSS. The demonstration is based on four interconnected software-defined radios to represent a four-satellite subset of a LEO constellation. Lab results show how each platform can generate a common timescale, with stability benefiting from all reference clocks.
1 INTRODUCTION
Constellations of small satellites in low Earth orbit (LEO) are currently being populated to provide communication (McDowell, 2020) and navigation (Prol et al., 2022; Reid et al., 2018) services. The global coverage provided by these proliferated LEO (pLEO) constellations presents an opportunity for alternative positioning, navigation, and timing (AltPNT) solutions that can augment existing global navigation satellite systems (GNSSs). Here, we briefly review oscillator-specific requirements in regards to navigation services provided from existing GNSSs and then discuss potential architectures for coordinating time and frequency across LEO platforms.
Systems that provide one-way time-of-flight-based navigation services require closely monitored, stable clocks from which ranging signals are generated. In the case of GNSSs, each satellite has multiple, highly stable, microwave atomic clocks integrated with a time-keeping system. By tracking GNSS signals from ground reference stations across the globe, the behavior of each clock is observed and ensembled together with laboratory-based clocks to produce a timescale, such as Global Positioning System time (GPST) (Coleman & Beard, 2020; Senior & Coleman, 2017) or Galileo time (Wang & Rochat, 2022), to which all member clocks are referenced. Corrections for each satellite clock with respect to the reference timescale are distributed to users via a navigation message, usually with an accuracy better than 10 ns (Bock et al., 2009).
A somewhat naïve approach would be to follow the GNSS model: putting independent, high-stability clocks onboard each satellite, monitoring them from the ground, and uploading clock corrections to each satellite for distribution to the users. This method is impractical for multiple reasons. First, because the clocks used on GNSS spacecraft are large and heavy, they are technologically and economically challenging to integrate onto smaller LEO spacecraft, which have much shorter orbital lifetimes. Additionally, because of the increased number of satellites required to provide global navigation service from LEO, observing each onboard clock in the pLEO constellation significantly increases tasking responsibilities for the ground monitoring network. The first challenge can be overcome by ensembling multiple low-size, weight, and power (SWaP) clocks (Flood et al., 2023; Van Buren et al., 2021) in place of a single high-performance clock; the second challenge requires a different strategy.
The availability of GNSSs in LEO provides a means for monitoring clocks directly onboard the AltPNT pLEO platform (Kunzi & Montenbruck, 2022; Kunzi et al., 2023). With suitable ephemeris knowledge, the LEO satellites could broadcast a one-way navigation signal. A drawback of this architecture is the dependence on continuous access to the GNSS: in the case of degradation or denial, the quality of service provided from pLEO would also be degraded.
In this paper, we explore a method that uses two-way time and frequency transfer between pLEO platforms to enable the use of low-SWaP clocks without dependence on large-scale ground monitoring or GNSS availability. The key is to exploit the extensive connectivity between the platforms already in place for their primary communication missions (Chaudhry & Yanikomeroglu, 2021; Harper, 2020; Heine et al., 2023). Radio frequency (RF) antennas and optical terminals are positioned on each spacecraft to enable in-plane and cross-plane connections; these cross-links can support data transfer, ranging measurements, and clock comparisons (Tournear, 2020). Each LEO satellite is assumed to have a small atomic clock onboard, and the existing RF cross-links enable clock comparison on different platforms, reducing the ground monitoring requirements. The inter-satellite clock measurements are input to established clock ensembling algorithms (Greenhall, 2006) to produce local realizations of pLEO time (pLEOT), a constellation timescale analogous to GPST. Because all members of the constellation are connected, traceability between pLEOT and coordinated universal time (UTC) can be provided either through connection with a small number of ground stations or via Global Positioning System (GPS) access by a subset of the pLEO constellation.
In this paper, we describe a small-scale RF signal demonstration of the proposed approach that makes three key contributions. First, we show a software-defined radio (SDR) representation of a small platform equipped with a low-SWaP atomic clock. The onboard clocks are compared by transmitting S band signals between the SDRs, producing differential phase measurements. Second, we demonstrate the implementation of a mixed clock ensemble using an algorithm that leverages unique clock stability profiles. Finally, another SDR realizes the timescale by using an ultra-low-noise oven-controlled crystal oscillator (OCXO) and numerically controlled oscillator (NCO) techniques.
The remainder of this paper is organized as follows: Section 2 shows how we represent a small satellite timing system using low-SWaP clocks and SDRs. Section 3 details the clock and measurement models used for estimating the clock states, with an emphasis on comparing two covariance reduction techniques used in clock ensembling algorithms. Section 4 describes how the ensemble mean is realized via NCO steering methods, which provide much better frequency control than directly steering an oscillator. The results demonstrate that precise clock phase measurements can be made at satellite communication frequencies, how these measurements can be used to ensemble clocks on separate platforms, and the synthesis of a clock signal disciplined to the ensemble.
2 HARDWARE OVERVIEW
In our laboratory, we use SDRs and low-SWaP atomic clocks to represent the timing system of a small satellite platform. The experimental block diagram in Figure 1 shows four SDR platforms, which represent four satellites in a single orbit plane. We assume that each platform has a direct RF communication channel to adjacent platforms, which is experimentally represented by RF cables; these connections enable precise clock phase measurements that are used in forming the clock ensemble. The implementation of the clock ensemble Kalman filter and the control system that realizes the implicit ensemble mean (IEM) are presented in Sections 3 and 4, respectively.
Multi-platform clock ensemble block diagram
(Top) Four interconnected platforms have small atomic clocks (green circles) as onboard oscillators, and 2-GHz signals (blue arrows) are synthesized from the onboard clocks and transmitted between platforms. (Bottom) The Kalman filters and signal generation components necessary for timescale formation are shown. Differential phase measurements (solid yellow arrows) derived from the S band transmissions are input to a clock ensemble Kalman filter. A differential phase measurement between the NCO signal (gray arrow) and a 10-MHz clock signal from Platform 4 (green arrow) is added to the phase estimate (pink arrow) for the clock on Platform 4; this quantity represents the phase offset between the NCO and the IEM estimate, which is driven to zero via feedback frequency adjustments (red arrow) output from the steering Kalman filter to the NCO. The performance of the IEM realization is evaluated by measuring the NCO with respect to a rubidium reference oscillator.
2.1 Software-Defined Radios
SDRs are primarily comprised of an analog RF front end, analog-to-digital converters, digital-to-analog converters (DACs), and a field-programmable gate array. Two different SDR models are used in this project, the Ettus N210 and the Ettus B210. The N210 is a networked device with two RF ports, external clock inputs, and modular daughterboards that each provide unique transmit and receive functionalities. In this project, we use the WBX transceiver daughterboard, which has a frequency range of 50–2200 MHz. The Ettus B210 is a single board transceiver with two transmit ports, two receive ports, and an external clock input. The frequency range of this device is 70 MHz to 6 GHz. Each platform shown in Figure 1 is realized by an SDR. The N210 SDRs, with one input and output, are used in Platform 1 and Platform 4, and the B210 SDRs, with two inputs and outputs, are used in Platform 2 and Platform 3. Each SDR transmits signals at 2 GHz, representing S band communication channels between platforms.
2.2 Clocks Onboard Small Platforms
In the following experiments, each SDR-based platform is equipped with a low-SWaP atomic clock connected to its external reference port. The clocks used are the Microsemi SA45.s chip scale atomic clock (CSAC) (Microsemi, 2019) and Safran mRO-50 (Safran, 2019). Both can feasibly be used on small satellite platforms as they are low-SWaP, relatively low-cost, commercially available in large quantities, and have models for space applications.
The most stable frequency reference in our laboratory is a Stanford Research Systems FS-725 rubidium frequency standard (Stanford Research Systems, 2019). Its stability was characterized at the National Institute of Standards and Technology in comparison to a hydrogen maser. All other clocks are measured against the rubidium standard via SDR-based clock characterization methods (Flood et al., 2023; Sherman & Jordens, 2016). The clock offsets are used to compute the overlapping Allan deviation (ADEV), shown in Figure 2. The profiles of all three CSACs have similar shapes, with the white frequency noise averaging down with a slope of τ−1/2 until the flicker floor is reached at a few thousand seconds. Each CSAC has slightly different performance, with CSAC 1 having the best stability and CSAC 3 having the worst stability. The mRO-50 has a stability curve that is shaped differently from any of the CSACs, with better stability than any CSAC at τ < 200 s. At τ > 600 s, the random walk noise processes in the mRO-50 result in a worse stability than any CSAC. A correctly implemented clock ensemble will take advantage of these complementary stability profiles, which should be apparent in the IEM realization.
Measured ADEV for clocks in the ensemble
The NEL OCXO has the best short-term stability and is used as the reference clock for the IEM realization. The mRO has the best medium-term stability (10 s < τ < 250 s), and CSAC 01 has the best long-term stability (τ >250 s). The ensemble realization will have hybrid stability properties primarily derived from the OCXO, mRO, and CSAC 01. All clocks are measured against the rubidium oscillator, which is only for laboratory verification and is not part of the proposed system.
An NEL Frequency Controls ultra-low-phase-noise OCXO (NEL Frequency Controls, 2019) was selected to drive the signal steering because of its exceptional short-term stability. The manufacturer specifications for the OCXO shown in Figure 2 are at or below the stability curve for the rubidium reference; thus, the true short-term behavior of the OCXO (τ < 4 s) cannot be measured in our system because of noise from the rubidium frequency reference. A proper assessment of the short-term OCXO noise would require a more stable reference clock for these averaging intervals.
2.3 Clock Phase Measurements
Previous work with the SDR testbed generated differential clock phase measurements directly from the 10-MHz output of each clock (Flood et al., 2023). In this experiment, the measurements are made based on 2-GHz signals transmitted between adjacent SDRs. The synthesized 2-GHz signal produced by each platform has the phase noise properties of its external oscillator, scaled by the ratio between the transmitted frequency and onboard clock frequency; in this case, the ratio is 200. Despite the absolute increase in phase noise on the synthesized carrier signal, the frequency stability of the synthesized signal matches that of the onboard oscillator from which it was derived, as many statistical phase noise quantities are normalized by the carrier frequency.
The phase of clock k is represented in Figure 1 as ϕk, and the differential phase measurements output from each platform are expressed as 
. The number of measurements made by each platform is equal to the number of platforms to which it is connected; Platforms 1 and 4 only make one differential phase measurement, whereas Platforms 2 and 3 each make two differential phase measurements. Phase measurements are processed in a central location for use in the clock ensemble Kalman filter. In an orbital scenario, this would require that all measurements be transmitted to a central processing location or distributed to every node.
3 CLOCK ENSEMBLING THEORY
The simplest definition of a clock ensemble is a group of clocks. In the absence of a more stable reference, as is the case in a deployed satellite system, measurements between group members can be used to estimate the phase and frequency of each clock with respect to some theoretical reference, called the IEM. These estimates could be applied as updates to each physical clock, providing a corrected signal that is more stable than the free-running physical clock (Greenhall, 2006). In most ensemble implementations, a separate signal source realizes the IEM. In the following section, we describe some of the steps required to perform the clock state estimation process, where the estimated state vector is defined as the offset of each true clock state with respect to the IEM: 
, where 
is the estimate error. The full details have been described by Flood et al. (2023); key points are reviewed here with an emphasis on comparing two ensembling algorithms.
3.1 Dynamic Model
We implement a simple two-state model (Zucca & Tavella, 2005) for each of the four clocks with the state vector shown in Equation (1). The oscillator states evolve according to the dynamics in Equation (2), with the state transition matrix detailed in Equation (3) and the process noise realizations represented by Equation (4). The discretization time step of the system is τ0:
1
2
3
4
The discrete process noise, wk, is assumed to be Gaussian with zero mean, and the covariance matrix Q(τ), shown in Equation (5), is described by white and random walk frequency noise parameters, q1,k and q2,k, specific to each clock, k. The computation of these noise parameters from ADEV values for an oscillator has been described by Zucca and Tavella (2005). The noise parameters in Table 1 were computed from the ADEV curve measured for each clock, as shown in Figure 2. Realizations of this noise, represented by the noise vector wk, are incorporated into simulated clock profiles according to Equation (2):
5
Measured Clock Noise Parameters
3.2 Measurement Model
The inputs to the filter are differential phase measurements between the ensemble members, with two measurements from each bidirectional RF link (δϕij, δϕji). The measurement input, zk, consists of six differential phase measurements at time step k with the measurement sensitivity matrix H shown in Equation (7). The phase difference measurement noise in our system was calibrated by using a common clock input to each receive port. The results were consistent for several clock inputs, including the CSACs, rubidium reference, and OCXO. The measurement noise for all systems is consistent with a zero-mean Gaussian with 1 – σ = 1 · 10−12 s. Thus, the measurement noise covariance (R) in the Kalman filter is set to an identity matrix scaled by σ2:
6
7
3.3 Covariance Reduction Methods
A system that estimates all member clock states using only differential clock phase measurements is unobservable, as an error common to all clocks does not appear in differential measurements (Brown, 1991). Covariance reduction techniques are required for unobservable systems because conventional Kalman filter implementations would result in unbounded covariance growth. Our previous work (Flood et al., 2023) used the covariance reduction algorithm originally developed by K. Brown to ensemble the GPS clocks (Brown, 1991). The Brown algorithm performs well when using clocks with similar stability; however, when ensembling clocks with diverse stability profiles, the IEM produced via Brown’s covariance reduction method does not properly capture the short-term stability of ensemble member clocks (Greenhall, 2011). An alternate covariance reduction method developed by C. Greenhall (Greenhall, 2006) at NASA’s Jet Propulsion Laboratory results in an ensemble average with better stability than all member clocks in the ensemble. These two covariance reduction methods are briefly described below.
3.3.1 Brown Covariance Reduction
The covariance reduction method detailed by Brown (1991) separates the observable and unobservable components of the state covariance matrix, 
, and uses the observable component to prevent unbounded uncertainty growth. The covariance reduction step defined in Equations (8)–(9) is implemented after the measurement update in the Kalman filter, and the resulting covariance matrix, Pr, is used as the covariance for the next time update at k + 1:
8
9
3.3.2 Greenhall Covariance Reduction
The requisite steps for the Greenhall covariance reduction algorithm (Greenhall, 2006) are shown in Equations (10)–(14). Notationally, 1n×m is a matrix of ones, 0n×m is a matrix of zeros, and In×n is an identity matrix. This method only uses the clock phase covariance elements from the full post-measurement update covariance matrix, P, arranged into a clock phase covariance matrix, Pϕ. A system of equations can then be expressed as a function of Equation (10) and a weight vector, w. The weight vector is constrained to sum to unity in Equation (11) and represents the weight of each clock in the ensemble average. The values for the clock weights computed from Equation (11) are summarized in Table 2. Equations (12)–(14) show the remaining steps for computing the reduced covariance, Pr; for full details, we refer the reader to the work by Greenhall (2006):
10
11
12
13
14
Clock Weights
3.3.3 Comparing Ensemble Algorithms
The two covariance reduction techniques were applied to the simulated clock ensemble to demonstrate the differences between the approaches. The clock ensemble was created by generating time series of simulated clocks using the process noise parameters in Table 1, making the differential phase measurements according to Equation (6), and using the conventional Kalman filter time and measurement update equations. Each covariance reduction approach was applied separately after the measurement update. The ADEV of the four simulated ensemble member clocks and the computed IEM for each ensemble algorithm are shown in Figure 3.
Simulated results from Brown and Greenhall algorithms
The two algorithms produce different results for the same input clock ensemble members. The IEM produced from the Brown algorithm only tracks CSAC 1 whereas the IEM produced from the Greenhall algorithm has a hybrid stability profile derived from both the mRO and CSAC 1.
For averaging intervals shorter than 2,000 s, the IEM computed via Brown’s covariance reduction follows CSAC 1, whereas the IEM computed using Greenhall’s covariance reduction follows the mRO. The ADEVs of the two IEM curves meet at an averaging interval of 2,000 s and agree for all longer averaging intervals. In this example, where the clock with the best stability changes as a function of the averaging interval, the choice of covariance reduction method impacts the ensemble output. In our previous work (Flood et al., 2023), this impact was not as clear, as the clocks all had similar stability profiles.
3.3.4 Clock Ensemble with S Band Measurements
When implementing a clock ensemble, care must be taken in processing signals at different frequencies. It is only appropriate to add or subtract phase measurements in units of radians if the signals have the same carrier frequency. However, measurements of phase converted to units of seconds allow for signal comparisons independent of carrier frequency. In our software implementation, the differential phase measurements (rad) made with the 2-GHz signals transmitted between SDR platforms are input to a custom processing block that handles conversion to seconds such that numerical precision is preserved within the Kalman filter, which operates in units of time. At the custom processing block output, the clock ensemble Kalman filter state estimates are converted back to radians, based on a 10-MHz carrier frequency. In Figure 1, a phase estimate from the Kalman filter, 
, is added to a differential phase measurement (rad), ϕNCO – ϕ4, which is based on 10-MHz signals. Throughout the experimental configuration in Figure 1, there are many phase conversions that occur; for a successful implementation, it is imperative to ensure that all conversions and scaling factors are correct.
4 NCO SIGNAL SYNTHESIS
IEM realization is achieved by steering an oscillator to the IEM estimate. In our previous work, clocks were steered by adjusting the applied voltage on a pin that changed the OCXO frequency (Flood et al., 2023). The performance of this approach was limited by the voltage source resolution and the responsiveness of the oscillator’s output frequency to changes in applied voltage. The computed control command had to be large enough to round to the nearest realizable frequency adjustment before the voltage was actually changed, resulting in frequency adjustment commands that were quantized at 41 μ Hz. With this method, performance improvements would require either voltage division at the DAC output or clocks with a flatter frequency response to an applied voltage, both of which would reduce the dynamic range of realizable frequency values for the output signal.
An alternative approach to IEM generation is using another SDR as an NCO. An NCO is a digital phase accumulator that increments by a specified phase step at a user-defined processing rate. The size of the phase step is determined by the desired signal frequency, f, and the sampling period of the NCO loop, Ts. At each loop iteration, the cumulative phase is incremented by the computed phase step. The signal frequency used to compute the NCO phase step size in our system is nominally 73 Hz, a prime number, which ensures signal sampling at unique locations. Trigonometric operations are performed on the accumulated phase to generate the output signal, which then goes through digital up-conversion to shift the 73-Hz signal to 10 MHz; the signal at the carrier frequency of 10 MHz is then sent to a DAC for transmission. The phase step and phase accumulation equations are shown in Equations (15)–(16):
15
16
The stability of signals generated by an unsteered NCO matches the stability of the oscillator from which the processing rate is derived. Thus, a reference oscillator with good short-term stability is desirable. In our experiment, long-term stability of the reference oscillator is less important, as the signal will be gently steered to the IEM. An NEL Frequency Controls, Inc. OCXO (NEL Frequency Controls, 2019) was used as the reference clock input to the SDR used as an NCO. Tests were conducted to confirm that the OCXO output and the signal synthesized by the NCO are of comparable stability.
4.1 NCO Steering
The repeated adjustment of oscillator frequency is known as clock steering. Clock steering techniques are used in the realization of timescales in order to take advantage of the best clock stability across different averaging intervals (Sherman et al., 2021). In this work, we use the same steering theory presented in previous work (Flood et al., 2023); however, the IEM is now realized as an NCO via Equations (15)–(16). The minimum realizable frequency adjustment is a function of the floating-point precision of the accumulated phase, the trigonometric operation on the phase, and the resolution of the DAC in the SDR. Because we are using double-precision floating-point variables to represent the phase over 0 to 2π, the decimal precision is approximately 1 · 10−15 rad. With this minimum phase precision and the sampling period of the NCO loop, Equation (15) can be used to solve for the minimum realizable frequency adjustment, which is 31 pHz. This approach improves the steering command precision with no reduction in the dynamic range of realizable frequency values.
In previous work (Flood et al., 2023), we evaluated the ability to steer an OCXO to a CSAC using an 18-bit DAC to change the applied voltage on a pin that changed the OCXO frequency. Here, we apply a similar test in which the NCO signal is steered to a CSAC. The CSAC in both of these experiments is the same clock; thus, any difference in signal performance can be attributed to the steering method. The goal of steering in both tests is to generate a signal with short-term stability from the OCXO and long-term stability from the CSAC. We evaluate steered signal performance in these simpler tests, where the phase offset of the NCO with respect to a target state is directly measurable. In a clock ensemble, the NCO phase offset from the IEM is an estimated quantity, thus coupling output signal stability with clock ensemble Kalman filter performance.
Figure 4 shows the ADEV of the target clock states and the two steered signals from the different experiments. The target state is defined by the same oscillator in both experiments; however, the actual behavior of the clock in any given experiment varies. Both steered signals were generated with a controller update rate of 1 Hz. The ADEV for the steered result using the original method with an 18-bit DAC is larger than the ADEV based on the NCO approach for all averaging intervals until these results meet at τ = 40 s. The ADEV for the steered result using the NCO approach preserves more of the free-running behavior of the OCXO until τ = 30 s, where it turns toward the target state. The NCO steering technique yields improved results because of the significant increase in steering command precision. The true short-term behavior of the NCO steered signal at τ < 4 s is obscured because of the noise of the rubidium frequency reference against which the signal is measured.
ADEV of the target state and steering results from separate experiments using two different steering methods
The steered result for Test 1 (solid red line) uses an 18-bit DAC to electronically steer the OCXO to the target state (dashed red line). The steered result for Test 2 (solid blue line) uses an NCO approach to steer the signal to the target state (dashed blue line). The steered result for Test 2 yields a better result than Test 1 because of the more precise resolution of the frequency adjustment commands.
4.1.1 Tuning NCO Signal Steering
One parameter used to tune the steered signal response is the system responsiveness to target state errors. Steering commands are based on a two-state error vector, representing the phase and frequency offset of the signal from the target state. The error vector, system dynamics, and pole placement control techniques (Brogan, 1991) are used to compute a gain matrix, G. The elements of G, which are indirectly designed via pole placement, will determine the responsiveness of the system.
The gain matrix, G, affects how quickly the signal transitions from the free-running behavior of the NCO reference clock to the IEM estimate. A larger gain will make the system more sensitive to error, which will produce larger frequency corrections for a given error vector; this can result in a more accurate realization of the IEM at intermediate averaging times, but comes at the cost of degraded short-term stability. Larger gain parameters are useful if the clock ensemble members have significantly better stability than the NCO reference clock. In contrast, smaller gain parameters will make the system less sensitive, producing smaller frequency corrections for a given error vector; this can preserve the short-term stability of the NCO reference clock, but can come at the cost of significant overshooting of the IEM.
Another tuning parameter is the controller update rate, which determines how much of the NCO reference clock behavior is preserved and changes the location of the servo bump (Fox et al., 2003), an effect that manifests in the steered signal stability because of the repeated application of a control input. Decreasing the control rate will preserve more of the NCO reference clock behavior and will shift the servo bump to longer averaging intervals. Increasing the control rate may result in a degradation of the short-term stability of the steered signal and will shift the servo bump to shorter averaging intervals. The optimal control frequency and gain parameters will vary depending on the application.
5 CLOCK ENSEMBLE TESTBED INTEGRATION
The formation of the multi-platform clock ensemble uses six SDRs, three CSACs, one mRO, one OCXO, and custom GNU Radio processing code. A subset of the block diagram from Figure 1 is shown in Figure 5. The interconnected SDRs represent four satellites in a single orbit plane. Each SDR has an onboard clock that is connected to the external reference port of the device; these four clocks constitute the members of the clock ensemble.
Simplified view of clock ensemble hardware showing the four platforms and the respective onboard clocks that form the clock ensemble
The four interconnected SDRs make differential clock phase measurements based on the 2-GHz signals that each device is transmitting and receiving. All of these measurements are sent via ethernet to a central location, where they are used in the clock ensemble Kalman filter to estimate the phase and frequency of the four ensemble member clocks. The mRO clock phase estimate is added to a differential phase measurement between the synthesized NCO signal and the mRO; this output is the phase offset between the synthesized NCO signal and the IEM. This phase offset is input to a simple steering filter, which produces a two-state error vector, minimized via feedback frequency adjustment commands to the NCO. The fifth SDR is the NCO signal source, and the sixth SDR makes measurements of the NCO, mRO, and rubidium frequency reference. The synthesized NCO signal is measured with respect to the rubidium frequency standard; evaluating the NCO signal in this manner would not be possible in a deployed system, as another high-stability reference would not be onboard.
5.1 Integrated Testbed Results
Multiple tests were conducted with different gain parameters and control rates to determine the optimal combination. The time series of the NCO signal phase was measured against the rubidium reference on the N310, and the ADEV was computed to assess signal stability. Results from the best configuration are shown in Figure 6 for a controller update rate of 4 Hz.
ADEV of the three CSACs (dashed blue) and mRO (solid blue), which are the four ensemble member clocks
The OCXO (gray) is the external reference clock for the NCO, which generates the IEM realization (red). The simulated IEM (black) from Figure 3 is used here as a benchmark to evaluate system performance.
There are three unique regions of stability for the IEM realization shown in Figure 6. The first region is for averaging intervals of τ < 5 s, where the steered signal closely matches the free-running OCXO ADEV; this result is caused by the NCO being driven by the free-running OCXO. The second stability region is for averaging intervals of 5 s < τ < 200 s, where the steered signal loosely follows the mRO ADEV. The NCO steered signal overshoots the mRO for averaging intervals of 5 s < τ < 70 s, because of the imperfect estimates of the ensemble clock states and the servo bump effect (Fox et al., 2003). The final stability region is for averaging intervals of τ > 200 s, where the steered signal follows the ADEV values of the most stable CSAC. The hybrid stability characteristics of the steered signal can be explained by the clock weights in Table 2: CSAC 1 and the mRO make up 94% of the weight in the clock ensemble whereas the other two clocks contribute 6%. If CSAC 2 and CSAC 3 were removed from the ensemble, the performance would likely be similar to what is shown here. However, in this multi-platform architecture, there is no detriment to including those two clocks in the ensemble. Despite a small ensemble weight, global distribution of either the measurements or clock estimates would enable any platform to realize the IEM, which would be pLEOT for the cluster.
The black ADEV curve in Figure 6 is the IEM computed from the simulation in Figure 3, where the simulated clocks were designed to have stability profiles that closely match the measured clock behavior. We can compare the IEM realization to the simulated IEM curve, which serves as the expected stability profile based on the simulated clock ensemble. The IEM realization closely matches the simulated IEM result, except for intermediate averaging intervals, where the IEM realization overshoots the mRO. Comparing the IEM realization to the expected curve based on simulation indicates that the system performs reasonably well, with performance improvements expected following reductions in the estimation error.
The red ADEV curve for the IEM realization in Figure 6 represents the best curve we have been able to produce. The definition of “optimal” will be application-specific. If the free-running stability of the NCO reference clock is significantly better than that of any ensemble members, a low control update rate and small gain parameters would most likely yield the best result. In contrast, if the free-running stability is lower than that of any ensemble members, a high control update rate and large gain parameters would likely yield the best result. In this work, the free-running stability of the NCO reference clock is only slightly better than the mRO; as a result, a moderate control frequency and moderate gain parameters were used to produce this output signal. If a reference clock with better short-term stability were available, the optimal control frequency and gain parameters would most likely change.
6 DISCUSSION
In this work, we demonstrated the ability to form a timescale by ensembling a diverse set of clocks driving separate SDRs. The differential phase measurements used in the ensemble were produced via unmodulated S band tones transmitted between devices. The realization of the IEM was achieved with a low-noise OCXO and NCO signal synthesis techniques. The synthesized NCO signal shows stability contributions from multiple clocks over averaging intervals for which their performance is best.
In this system architecture, each atomic clock is free-running, and the inter-platform measurements are sent to a single platform for processing. Because Clock 4 is used for IEM estimation in Figure 1, this experimental implementation demonstrates the requisite steps for timescale formation onboard Platform 4. Other platforms could use the same methods to generate onboard realizations of pLEOT if the inter-platform measurements are distributed to all participating nodes. All platforms will attempt to generate the same realization of pLEOT, each using their own local reference clock and low-noise oscillator.
Alternatively, clock estimates from a clock ensemble Kalman filter running onboard a single platform can be distributed to every participating node in the network. With an estimate of its local clock, each platform could realize pLEOT by using the techniques described above without the overhead of a Kalman filter running on each node. Research into the optimal architecture for each platform to access or generate pLEOT is in progress.
In principle, because the proposed pLEO constellations will be fully interconnected, the methods described above could be used to increase system autonomy by forming a timescale based primarily on inter-satellite measurements. If every platform creates a realization of pLEOT, each onboard navigation system can generate signals directly referenced to pLEOT. Because all of the platforms are interconnected, a minimum of one connection to either GPS or a single ground station would be required to compute the offset of pLEOT with respect to GPST or UTC.
7 CONCLUSIONS & FUTURE WORK
The hardware architecture described here enables clock ensemble formation using atomic clocks located on separate platforms. The timescale realized from this experimental configuration has better stability than any individual member of the ensemble for almost all averaging intervals. These methods provide an initial step towards architectures that support timescale formation across a pLEO constellation that could provide navigation services.
Future work will analyze the scalability of the experimental methods in the current centralized processing architecture and explore decentralized architectures as potential alternatives. Additional analysis will be conducted regarding the impact of platform dynamics and relativistic effects on clock measurements. First-order Doppler effects will change the frequency of received signals as a function of relative velocity between platforms. Second-order Doppler effects and differences in gravitational potential will affect oscillator frequency as observed from another platform, ultimately manifesting as offsets in carrier frequency on received signals. Future studies will incorporate these effects and assess their impact on timescale formation within a pLEO constellation.
HOW TO CITE THIS ARTICLE
Flood, C., & Axelrad, P. (2024). Timescale realization with linked platforms for altPNT. NAVIGATION, 71(4). https://doi.org/10.33012/navi.669
ACKNOWLEDGMENTS
We gratefully acknowledge the contributions to this work made by Justin Pedersen, which include modeling of clocks onboard the SDRs and assistance in implementing the timescale formation experiment.
This work was supported by Air Force Research Laboratory (AFRL) grant FA9453-19-1-0076 to the University of Colorado Boulder and a subaward to the University of Colorado Boulder on an AFRL award FA9453-20-2-0001 to the University of New Mexico.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the United States Air Force. The appearance of external hyperlinks does not constitute endorsement by the United States Department of Defense (DoD) of the linked websites or the information, products, or services contained therein. The DoD does not exercise any editorial, security, or other control over the information you may find at these locations.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Jump to section
Related Articles
Cited By...
- No citing articles found.