Quantifying the force regime that controls the movement of a single grain during fluvial transport has historically proven to be difficult. Inertial micro-electromechanical system (MEMS) sensors (sensor assemblies that mainly comprise micro-accelerometers and gyroscopes) can used to address this problem using a “smart pebble”: a mobile inertial measurement unit (IMU) enclosed in a stone-like assembly that can measure directly the forces on a particle during sediment transport. Previous research has demonstrated that measurements using MEMS sensors can be used to calculate the dynamics of single grains over short time periods, despite limitations in the accuracy of the MEMS sensors that have been used to date. This paper develops a theoretical framework for calculating drag and lift forces on grains based on IMU measurements. IMUs were embedded a spherical and an ellipsoidal grain and used in flume experiments in which flow was increased until the grain moved. Acceleration measurements along three orthogonal directions were then processed to calculate the threshold force for entrainment, resulting in a statistical approximation of inertial impulse thresholds for both the lift and drag components of grain inertial dynamics. The ellipsoid IMU was also deployed in a series of experiments in a steep stream (Erlenbach, Switzerland). The inertial dynamics from both sets of experiments provide direct measurement of the resultant forces on sediment particles during transport, which quantifies (a) the effect of grain shape and (b) the effect of varied-intensity hydraulic forcing on the motion of coarse sediment grains during bedload transport. Lift impulses exert a significant control on the motion of the ellipsoid across hydraulic regimes, despite the occurrence of higher-magnitude and longer-duration drag impulses. The first-order statistical generalisation of the results suggests that the kinetics of the ellipsoid are characterised by low- or no-mobility states and that the majority of mobility states are controlled by lift impulses.

River sediment transport is a critical process in landscape evolution

Fluvial sediment transport is a complex two-phase flow defined by (a) hydraulics

To analyse the motion of a grain resting on a riverbed that is sheared by a turbulent flow

Field experiments tracing individual sediment grains are in principle Lagrangian

Lagrangian measurements find direct application in coarse-grain gravel bed and bedrock river environments (e.g.

A particular advance in monitoring technology has been the development of sediment grain scale inertial sensors which record at high frequency the accelerations and angular velocities experienced by grains during entrainment and motion

MEMS-IMU sensors measure the acceleration and angular velocity of the grain, which can be used to calculate the net force acting on the grain. For the most accurate measurement of this force, sensors should be located at the grain's centre of mass. Data collected from within grains undergoing transport has potential to describe the timing of motion, forces acting on the grain and grain location. As a grain moves, its centre of mass moves, and so the reference point for the force measurements is mobile. The latter means that the IMU measurements need to be transformed to a frame of reference that can be understood by an observer. Generally, an IMU accelerometer is a non-inertial frame fixed within the mobile body frame of the sensor assembly.

In theory, the accelerations recorded by the IMU could be integrated to calculate grain velocity and integrated again to reveal location, a process referred to as dead reckoning. One long-term goal for this approach would be to use IMUs to track a large number of grains through fluvial systems. However, real fixed (“strap-down”) IMUs based on MEMS are not suitable for these integrations since the data contain several sources of uncertainty, including signal noise and nanoscale misalignment of sensor axes. With sensors that are cheap enough to be deployed in large numbers, the accumulation of errors means that they cannot be used for 3D tracking of long-term unconstrained motions

Despite the limitations, IMU sensors have been considered to be a suitable technique for measuring grain motion (e.g.

The first goal of this paper is to introduce a simple rigid-body model that connects measurements derived from an idealised IMU with existing models for grain motion. For this model to be successful, it is necessary to map the IMU body frame dynamics to the reference frame of motion (flume or riverbed). This resolution allows the inertial measurements to be related to the forcing on the particle and defines an explicit and unambiguous threshold for particle motion. The second goal is to introduce the calculation of inertial impulses above the drag and lift thresholds of motion, following the example of

An IMU accelerometer records accelerations of the grain, which can be converted to forces acting on the grain by multiplying by grain mass. However, those accelerations are both fixed (e.g. due to gravity) and variable (e.g. due to fluid forces applied to the particle). The measurements are difficult to interpret because fixed and variable accelerations cannot be decoupled after the accelerometer begins to move, and these accelerations are perceived differently from different reference frames. If the sensor is static and gravity is recorded from the reference frame of the sensor, it is then possible to derive a reliable orientation measurement. If the sensor moves and the sensor frame accelerations are compensated for gravity (by removing gravity from the raw accelerometer measurement), then the sensor will record the 3D components of the resultant or net force that mobilises the particle. This resultant is the force that can be observed from an observer who is static in relation to the particle. In Appendix

The physics of IMU sensors also define the main difference between IMUs and other force sensors that have been used to monitor grain motion, such as load cells (e.g.

The following derivations rely on the transformation of the IMU accelerations from the mobile reference frame of the monitored particle (frame

Forces in the Newton–Euler model. The diagram shows the linear forces applied on the centre of the mass of the target particle.

Combined measurements of grain acceleration and angular velocity allow direct calculation of the forces and turning moments acting at the grain's centre of mass. This type of model formulation is the Newton–Euler model in the rigid-body-dynamics literature

The left-hand elements of Eqs. (

The magnitude of

For the direction normal to the bed (

The linear force threshold of motion is defined for

If

For the lift direction, the component of

The rotational threshold of motion is defined by the state where the balance of torques around the centre of mass of the particle are balanced (

The torques are analysed in the body frame of the particle (

It is useful to re-state here that all the derivations so far are for a spherical particle (the equivalent of Eq. (

For completeness we note that, if

To account for both the duration and the magnitude of a force, the impulse

Two sensor assemblies were deployed, one sphere and one ellipsoid (described in

All flume experiments took place in an Armfield

Laboratory setting and initial alignment. The diagram shows the arrangement of the bed of hemispheres and the test position (4.5 m downstream of the entrance of the flume). Upstream of the bed of hemispheres, the flume was filled with densely packed gravel (

For the experiments, the spherical device was placed on the flume centreline in a saddle position between four bed hemispheres, and the three sensor axes were aligned with the inertial frame

Ten entrainment experiments were conducted with each device. We define entrainment as when the particle moves by one particle diameter or

Example flume and field experiments

Entrainment was observed independently from video recordings which were synchronised with the experiments from the start of the flow increase (Sect.

Field experiments took place within a 5 m long straight and confined reach of the Erlenbach mountain stream in Switzerland, approximately 15 m upstream of the concrete channel section and 55 m upstream of the sediment retention basin in which continuous bedload transport measurements have been made during the past 30 years

The flume experiments demonstrate the differences between the spherical and the ellipsoid particle during incipient motion (Figs.

Inertial impulses and duration of threshold exceedance events for laboratory experiments. Impulses of all the inertial forces that exceeded the gravity forces. For the spherical particle

Probabilistic inertial impulse threshold for laboratory experiments. Logistic regression of the probability of entrainment for the spherical

The results from the ellipsoid sensor demonstrate a strong influence of the lift forces. Exceedance impulses occur for similar durations and magnitudes; however there is a strong bias of the lift distribution towards the shorter and low-impulse events. The drag duration and impulse distributions include more and higher magnitude outliers than the lift distributions (

Using the video recording observations, the impulse thresholds for entrainment were approximated with logistic regression. The probability of 0.5 corresponds to the threshold impulse for which the probability changes from the particle being more likely to be at rest to being more likely to be entrained. In this context, with this approximation we calculate a gradational threshold of entrainment

Finally, the results from the field experiments (Fig.

Inertial Impulses and duration of threshold exceedance events for field experiments. Impulses of all the inertial forces that exceeded the gravity forces. During short-transport events (average travel distance

The advantage of using an IMU sensor for capturing grain motion is that the sensor solves a complex force and torque balance and removes any ambiguity in whether or not a test particle is in motion, as motion leads to the explicit thresholds

Further, there has been a recent rapid increase in use of IMU sensors, but most off-the-shelf IMU sensors are not suitable for the range of forces characterising natural sediment transport, especially if the focus is on particle interaction or impacts

Here, we calibrated and deployed a commercial IMU sensor following standard procedures

Two aspects of this study are particularly important to address before we make comparisons with previous studies. The first is that we made inertial measurements from within the sediment particles, which are fundamentally different from measurements of fluid turbulence that are often used for predicting sediment motion. The second is that the flow regimes under which we made measurements, with varying shallow flows, differ from those in many studies of sediment motion. Both of these aspects provide new insights into sediment movement, but they require care in making direct comparisons with studies that have used different approaches and/or hydraulic conditions. In addition, it is useful to note that grain protrusion is not discussed in this work, despite being an important control on grain motion and particularly entrainment

This work uses a theoretical framework which has the potential to enhance the mathematical modelling of sediment transport. The Newton–Euler model of Sect.

Previous laboratory studies using fixed force meters attached to grains

Static vibration sensors were also deployed by

The laboratory inertial impulse calculations demonstrate that, for unrestricted entrainments, there are observable differences between spherical and ellipsoid particles, with the latter being more sensitive to the lift forces at entrainment threshold conditions. Those differences support previous work on the effect of shape on the response of particles in various hydraulic regimes (e.g.

The corresponding inertial calculations from the field also demonstrate that the ellipsoid is highly sensitive to lift forces and impulses. We observe higher mean magnitude lift forces on the ellipsoid in the field (2.57 N) than in the laboratory (2.13 N) (Fig.

Overall, differences of particle inertial dynamics during grain entrainment and translation are important because they can potentially enhance predictions for grain particle travel distances with measurements from the field and particularly for large distances

Considerable effort has been applied to define distributions for hydraulic impulses during the entrainment of spherical particles and relate them to critical thresholds

Our data provide new insights into the roles of drag and lift impulses in entrainment. To begin to generalise these findings and to assess the interactions between lift and drag forces, a bootstrapping method is used here. We approximate the distributions of inertial lift and drag impulses for an ellipsoid particle and from a combination of laboratory and field measurements. This analysis is also the first step towards calculating the combined behaviour of the drag and lift distributions, which can lead to the definition of joint distributions that have stronger explanatory and perhaps predictive value.

To combine the results from the laboratory and field ellipsoid experiments, the impulsive exceedance events were normalised since the conditions are different for the laboratory and the natural conditions. Normalisation used the mean impulses for all drag and all lift forces. Also, the mean impulses were calculated separately for the laboratory and field experiments (

Bootstrap normalised impulse sampling (lift and drag).

The fitting of the representative distributions for

This work introduces a framework that can be used to derive and interpret IMU measurements in sediment transport studies. The derivation of inertial measurements from mobile sediment grains requires a physical model that links the inertial dynamics with existing force (or moment) balance equations for sediment transport. The types of sensors and associated smart-pebble assemblies currently deployed for the measurements of grain inertial dynamics are not suitable for 2D or 3D tracking of grain position. However, it is possible to measure net forces and impulses if the necessary transformations are applied consistently.

Field and laboratory measurements of inertial lift and drag impulses highlight the different entrainment behaviours of a spherical and an ellipsoidal particle. The lift net force is dominant during the unrestricted entrainment of the ellipsoid, while there is no statistical difference between the effects of lift and drag impulses on the entrainment of the sphere. The drag component can be stronger during transport; however short impulses influence the motion of the ellipsoid significantly.

The continuous improvement of the sensor technology along with the better understanding of the physics described by inertial measurements can lead to a unified treatment of the resultant grain dynamics during bedload transport. These are the dynamics that represent exactly the interaction of hydraulic and sediment forces in different regimes and can enhance the parametrisation of important hydro-morphological controls.

To discuss the measurements recorded by an IMU, and particularly the measurements from an accelerometer and a gyroscope, it is necessary to introduce three basic frames of reference and select one of the many representations for arbitrary rotations in 3D.

The following assumptions and simplifications are used throughout this study:

Due to the small-scale (10

For the same reason, the non-gravitational fictitious forces (such as the Coriolis effect) are ignored.

For the mathematical derivations, ideal IMUs (no error accumulation is considered) and perfectly aligned sensor assemblies are assumed. The errors associated with IMUs and especially with the magnitude of the integration errors are presented in relevant electrical engineering sources (e.g.

We define the body frame

The local geographical frame

Frames of reference.

Transforming information between these three reference frames is non-trivial. A widely used method to represent the change between frame is the application of quaternions

A unit quaternion

Equation (

Inertial accelerometers measure the proper acceleration

In Eq. (

Example incipient motion IMU data. Measurements from the incipient motion experiments using the spherical sensor.

The rotational component captured by the gyroscope relates to the moments of the forces applied on the surface of the measured particle via Eq. (

For the ellipsoid, the same calculation is significantly more complicated. Firstly, the moment of inertia is not uniform. In this work, we implemented a numerical calculation during the design phase of the enclosure (using Solidworks,

For the calculation of the tangential force, a good approximation (ignoring the secondary moments of inertia) is given by dividing the principle moments by the half length of the corresponding principle axis of the ellipsoid, giving

Force magnitude of the rotational component.

The parameters are estimated as follows:

Histogram of inertial forces from all experiments. The inertial dynamics show that net lift (

Lift vs. drag force magnitude correlation (flume experiments). Regression analysis applied to the magnitude of calculated forces (drag and lift) shows a moderate correlation for the spherical particle (statistically significant Pearson's

Three types of right-tail distributions were considered (

Figure

Figure

Choice of distribution for drag impulses.

Choice of distribution for lift impulses.

Fitted distribution statistics.

Statistics for selected distributions.

Bootstrap parameters for selected distributions. The graphs quantify the stability of the selected distributions for drag and lift impulses. For the gamma distribution (drag impulses) 1000 bootstrapped parameters were cross-compared, revealing a range of 0.1 for the shape parameter and 0.2 for the rate parameter

Quaternions can be written in the form

The quaternion conjugate is given by

The sum of two quaternions is then

The quaternion norm is therefore defined by

Quaternions can be interpreted as a scalar plus a vector by writing

Finally, the rotation about the unit vector

The components of this quaternion are called Euler parameters. After rotation, a point

A concatenation of two rotations, first

Finally, the transformation that gives the equivalent DCM for a quaternion

To demonstrate the advantage of quaternions, we randomly rotate the static vector of gravity. In an orthogonal Cartesian frame where the origin of the

Randomisation of the body frame angular velocities of the rigid body

Calculation of successive quaternions using direct multiplication for random angular velocities.

Calculation of the direction cosine matrix from Euler angles.

Rotating the vector of gravity in the body frame of the rigid body using both the direction cosine matrix and common matrix vector multiplication.

Random rotation of the static vector of gravity.

The dataset used in this paper is curated by Georgios Maniatis in the following Github repository:

All the calculations were performed using the R statistical software and specifically the libraries orientlib

GM designed and calibrated the sensor, designed and implemented the experiments, performed all the physical and statistical calculations, and produced the first draft of this paper. TH supervised the laboratory experiments, reviewed several versions of the manuscript, and contributed significantly to the interpretation and contextualisation of the results. RH contributed to the design of the laboratory experiments, reviewed several versions of the manuscript, and contributed to the interpretation and contextualisation of the results. DR contributed to the design, supervised and assisted with the field experiments, reviewed several versions of the manuscript, and contributed to the interpretation and contextualisation of the results. AB contributed to the design of the field experiments and reviewed several versions of the manuscript.

The authors declare that they have no conflict of interest.

The flume experiments were conducted in the School of Engineering of the University of Glasgow. The authors thank Tim Montgomery (University of Glasgow) and Tobias Nicollier (WSL) for their assistance with the laboratory and field experiments, respectively; three anonymous reviewers, who significantly improved the manuscript with their contribution; and Jens Turowski, who provided feedback on previous versions of the manuscript. Finally, Georgios Maniatis thanks Katerina Georgiou for her assistance with the design and production of the figures and the typewriting of the manuscript.

Georgios Maniatis was supported by a University of Glasgow Lord Kelvin Adam Smith Scholarship (charity number SC004401) at the time of the flume experiments. The field experiments were supported by an Early Career Research award from the British Society for Geomorphology (Quantification of erosional processes using inertial sensors across environments, 2017).

This paper was edited by Claire Masteller and reviewed by three anonymous referees.