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T. Visser
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1
The CHAMP and GOCE satellites provided high-resolution thermosphere data between 2000 and 2013, improving our knowledge of atmosphere dynamics in the thermosphere-ionosphere region. However, the currently available data sets contain inconsistencies with each other and with external data sets and models, arising to a large extent from errors in the modelling of aerodynamic forces. Improved processing of the wind data for the two satellites would benefit the further development and validation of thermosphere models and improve current understanding of atmospheric dynamics and long-term trends. The first step to remove inconsistencies has been the development of high-fidelity models of the satellite surface geometry. Next, an improved characterization of the collisions between atmospheric particles and satellite surfaces is necessary. In this article, the effect of varying the energy accommodation coefficient, which is a key parameter for describing gas-surface interactions (GSI) is investigated. For past versions of the thermosphere density and wind data from these satellites a value of the energy accommodation coefficient of αE=0.93 was selected. The satellite accelerometer measurements, from which the thermospheric data are derived, have now been reprocessed using high-fidelity geometries and a wide range of αE values. Lowering the αE value used in the processing leads to an increase in the lift over drag ratio for those satellite panels that are inclined to the flow. This changes the direction of the modelled acceleration, and therefore the interpretation of the measured acceleration in terms of wind. The wrong choice of αE therefore leads to the introduction of satellite attitude-dependent wind errors. For the CHAMP and GOCE satellites, we have found that values of the energy accommodation coefficient significantly lower than 0.93 (0.85 for CHAMP and 0.82 for GOCE) result in increased consistency of the wind data. A comparison between the two missions and an overview of the influence on the results of filtering for solar activity and seasonal and diurnal variations is presented.
...
The CHAMP and GOCE satellites provided high-resolution thermosphere data between 2000 and 2013, improving our knowledge of atmosphere dynamics in the thermosphere-ionosphere region. However, the currently available data sets contain inconsistencies with each other and with external data sets and models, arising to a large extent from errors in the modelling of aerodynamic forces. Improved processing of the wind data for the two satellites would benefit the further development and validation of thermosphere models and improve current understanding of atmospheric dynamics and long-term trends. The first step to remove inconsistencies has been the development of high-fidelity models of the satellite surface geometry. Next, an improved characterization of the collisions between atmospheric particles and satellite surfaces is necessary. In this article, the effect of varying the energy accommodation coefficient, which is a key parameter for describing gas-surface interactions (GSI) is investigated. For past versions of the thermosphere density and wind data from these satellites a value of the energy accommodation coefficient of αE=0.93 was selected. The satellite accelerometer measurements, from which the thermospheric data are derived, have now been reprocessed using high-fidelity geometries and a wide range of αE values. Lowering the αE value used in the processing leads to an increase in the lift over drag ratio for those satellite panels that are inclined to the flow. This changes the direction of the modelled acceleration, and therefore the interpretation of the measured acceleration in terms of wind. The wrong choice of αE therefore leads to the introduction of satellite attitude-dependent wind errors. For the CHAMP and GOCE satellites, we have found that values of the energy accommodation coefficient significantly lower than 0.93 (0.85 for CHAMP and 0.82 for GOCE) result in increased consistency of the wind data. A comparison between the two missions and an overview of the influence on the results of filtering for solar activity and seasonal and diurnal variations is presented.
Recently, the horizontal and vertical cross wind at 225- to 295-km altitude were derived from linear acceleration measurements of the Gravity field and steady-state Ocean Circulation Explorer satellite. The vertical component of these wind data is compared to wind data derived from the mass spectrometers of the Atmosphere Explorer C and E and Dynamics Explorer 2 satellites. From a statistical analysis of the 120-s moving-window standard deviation of the vertical wind (σ(Vz)), no consistent discrepancy is found between the accelerometer-derived and the mass spectrometer-derived data. The validated Gravity field and steady-state Ocean Circulation Explorer data are then used to investigate the influence of several parameters and indices on the vertical wind activity. To this end, the probability distribution of σ(Vz) is plotted after distributing the data over bins of the parameter under investigation. The vertical wind is found to respond strongly to geomagnetic activity at high latitudes, although the response settles around a maximum standard deviation of 50 m/s at an Auroral Electrojet index of 800. The dependence on magnetic local time changes with magnetic latitude, peaking around 4:30 over the polar cap and around 01:30 and 13:30 in the auroral oval. Seasonal effects only become visible at low to middle latitudes, revealing a peak wind variability in both local summer and winter. The vertical wind is not affected by the solar activity level.
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Recently, the horizontal and vertical cross wind at 225- to 295-km altitude were derived from linear acceleration measurements of the Gravity field and steady-state Ocean Circulation Explorer satellite. The vertical component of these wind data is compared to wind data derived from the mass spectrometers of the Atmosphere Explorer C and E and Dynamics Explorer 2 satellites. From a statistical analysis of the 120-s moving-window standard deviation of the vertical wind (σ(Vz)), no consistent discrepancy is found between the accelerometer-derived and the mass spectrometer-derived data. The validated Gravity field and steady-state Ocean Circulation Explorer data are then used to investigate the influence of several parameters and indices on the vertical wind activity. To this end, the probability distribution of σ(Vz) is plotted after distributing the data over bins of the parameter under investigation. The vertical wind is found to respond strongly to geomagnetic activity at high latitudes, although the response settles around a maximum standard deviation of 50 m/s at an Auroral Electrojet index of 800. The dependence on magnetic local time changes with magnetic latitude, peaking around 4:30 over the polar cap and around 01:30 and 13:30 in the auroral oval. Seasonal effects only become visible at low to middle latitudes, revealing a peak wind variability in both local summer and winter. The vertical wind is not affected by the solar activity level.
Thermospheric wind measurements obtained from linear non-gravitational accelerations of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite show discrepancies when compared to ground-based measurements. In this paper the cross-wind is derived from both the linear and the angular accelerations using a newly developed iterative algorithm. The two resulting data sets are compared to test the validity of wind derived from angular accelerations and quantify the uncertainty in accelerometer-derived wind data. In general the difference is found to be less than 50 m/s vertically after high-pass filtering, and 100 m/s horizontally. A sensitivity analysis reveals that continuous thrusting is a major source of uncertainty in the torque-derived wind, as are the magnetic properties of the satellite. The energy accommodation coefficient is identified as a particularly promising parameter for improving the consistency of thermospheric cross-wind data sets in the future. The algorithm may be applied to obtain density and cross-wind from other satellite missions that lack accelerometer data, provided the attitude and orbit are known with sufficient accuracy.
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Thermospheric wind measurements obtained from linear non-gravitational accelerations of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite show discrepancies when compared to ground-based measurements. In this paper the cross-wind is derived from both the linear and the angular accelerations using a newly developed iterative algorithm. The two resulting data sets are compared to test the validity of wind derived from angular accelerations and quantify the uncertainty in accelerometer-derived wind data. In general the difference is found to be less than 50 m/s vertically after high-pass filtering, and 100 m/s horizontally. A sensitivity analysis reveals that continuous thrusting is a major source of uncertainty in the torque-derived wind, as are the magnetic properties of the satellite. The energy accommodation coefficient is identified as a particularly promising parameter for improving the consistency of thermospheric cross-wind data sets in the future. The algorithm may be applied to obtain density and cross-wind from other satellite missions that lack accelerometer data, provided the attitude and orbit are known with sufficient accuracy.
Cross-wind from linear and angular satellite dynamics
The GOCE perspective on horizontal and vertical wind in the thermosphere
The decay of satellite orbits has been used extensively to obtain thermospheric density measurements. With the introduction of accelerometers in spacecraft, the spatial resolution of these data could be increased. At the same time, the direction of the measured acceleration provides a measure for the direction of the incoming flow, and therefore of the local cross-wind. In this thesis, the angular acceleration of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, an Earth explorer by the European Space Agency (ESA), is used as a source for such thermospheric wind data for the first time. The goal is to improve aerodynamic parameter estimates and assess the quality of accelerometer-derived wind data by comparing this new data set to that derived from linear accelerations...
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The decay of satellite orbits has been used extensively to obtain thermospheric density measurements. With the introduction of accelerometers in spacecraft, the spatial resolution of these data could be increased. At the same time, the direction of the measured acceleration provides a measure for the direction of the incoming flow, and therefore of the local cross-wind. In this thesis, the angular acceleration of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, an Earth explorer by the European Space Agency (ESA), is used as a source for such thermospheric wind data for the first time. The goal is to improve aerodynamic parameter estimates and assess the quality of accelerometer-derived wind data by comparing this new data set to that derived from linear accelerations...
High performance control allocation methods for the Innovative Control Effectors (ICE) aircraft require accurate onboard aerodynamic models, with preferably first order continuity. Simplotope B-Splines, an extension on Simplex B-Splines, have a high approximation power by using local cost functions. However, enforcing global continuity produces computationally expensive optimization problems. This paper presents a distributed approach, using the Alternating DirectionMethod of Multipliers (ADMM), to reduce the complexity of the B-Coefficients’ estimation. ADMM decouples the simplotopes, and introduces coupling coefficients to enforce global continuity, resulting in a parallel estimation algorithm whose complexity is depending solely on the partition size, being independent of refinement of the model tessellation. Results show that for a 3D model, the distributed algorithm converges steadily to the global solution with a good approximation after a couple hundred iterations. Validation results of the distributed approach were similar to those of the global optimal solution for various noise intensities, and the continuity constraints were satisfied with maximum mismatches below 10-4. The distributed approach has been used to construct a first order continuous aerodynamic model for the ICE aircraft, which has been implemented in Simulink, and proven to perform well compared to the original model.
...
High performance control allocation methods for the Innovative Control Effectors (ICE) aircraft require accurate onboard aerodynamic models, with preferably first order continuity. Simplotope B-Splines, an extension on Simplex B-Splines, have a high approximation power by using local cost functions. However, enforcing global continuity produces computationally expensive optimization problems. This paper presents a distributed approach, using the Alternating DirectionMethod of Multipliers (ADMM), to reduce the complexity of the B-Coefficients’ estimation. ADMM decouples the simplotopes, and introduces coupling coefficients to enforce global continuity, resulting in a parallel estimation algorithm whose complexity is depending solely on the partition size, being independent of refinement of the model tessellation. Results show that for a 3D model, the distributed algorithm converges steadily to the global solution with a good approximation after a couple hundred iterations. Validation results of the distributed approach were similar to those of the global optimal solution for various noise intensities, and the continuity constraints were satisfied with maximum mismatches below 10-4. The distributed approach has been used to construct a first order continuous aerodynamic model for the ICE aircraft, which has been implemented in Simulink, and proven to perform well compared to the original model.
Abstract
(2018)
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Eelco Doornbos, C Stolle, Martin Rother, Lu Gang, Brian Anderson, Tim Visser, Jose van den IJssel, Pieter Visser, Bjorn Frommknecht, Fabrice Cipriani
Satellite gravity field missions, such as CHAMP, GRACE, GOCE and GRACE-FO, are designed to make extremely accurate measurements in low Earth orbit of the (relative) motion of satellites, and proof masses inside these satellites, in order to infer information on the Earth’s gravity field. There are several mechanisms through which the thermosphere and ionosphere influence the way in which this motion is measured. The accelerometers, drag-free control system, dual-frequency radio-tracking systems, star cameras and magnetometers that are part of the payloads and platforms of the satellites, are to some extent also devices for active space weather monitoring. The space weather observations, which are considered an error source to be mitigated for the primary goal of gravity field determination, are a valuable source of information for studies of the thermosphere-ionosphere. For nearly two decades now, data from the accelerometers on board these gravity field missions have had a very large impact on studies of the thermosphere density and horizontal neutral wind. Accurate information on ionospheric electron content has been derived from dual-frequency satellite-to-satellite tracking systems as well. More recent developments are the derivation of vertical neutral wind from GOCE acceleration data and the calibration of platform magnetometer data, converted into ionospheric field-aligned current densities. Using these processing techniques, GOCE, GRACE and GRACE-FO can be turned into sources of valuable thermosphere-ionosphere data, that supplement dedicated missions such as Swarm, in terms of temporal and spatial coverage. To demonstrate the value of this data in the space weather and space physics domains, we provide comparisons of these data with other data sources, such as AMPERE, as well as with output of a global thermosphere-ionosphere general circulation model. It is noteworthy that satellite mission concepts for investigations of thermosphereionosphere coupling, that are currently under study, would carry similar devices and benefit from similar satellite platform designs and orbits as the current gravity field missions. It is clear that there is a large potential to increase this synergy today, and opportunity for dual-purpose missions in the future. We therefore conclude by providing recommendations in terms of instrumentation and data handling for present and future missions.
...
Satellite gravity field missions, such as CHAMP, GRACE, GOCE and GRACE-FO, are designed to make extremely accurate measurements in low Earth orbit of the (relative) motion of satellites, and proof masses inside these satellites, in order to infer information on the Earth’s gravity field. There are several mechanisms through which the thermosphere and ionosphere influence the way in which this motion is measured. The accelerometers, drag-free control system, dual-frequency radio-tracking systems, star cameras and magnetometers that are part of the payloads and platforms of the satellites, are to some extent also devices for active space weather monitoring. The space weather observations, which are considered an error source to be mitigated for the primary goal of gravity field determination, are a valuable source of information for studies of the thermosphere-ionosphere. For nearly two decades now, data from the accelerometers on board these gravity field missions have had a very large impact on studies of the thermosphere density and horizontal neutral wind. Accurate information on ionospheric electron content has been derived from dual-frequency satellite-to-satellite tracking systems as well. More recent developments are the derivation of vertical neutral wind from GOCE acceleration data and the calibration of platform magnetometer data, converted into ionospheric field-aligned current densities. Using these processing techniques, GOCE, GRACE and GRACE-FO can be turned into sources of valuable thermosphere-ionosphere data, that supplement dedicated missions such as Swarm, in terms of temporal and spatial coverage. To demonstrate the value of this data in the space weather and space physics domains, we provide comparisons of these data with other data sources, such as AMPERE, as well as with output of a global thermosphere-ionosphere general circulation model. It is noteworthy that satellite mission concepts for investigations of thermosphereionosphere coupling, that are currently under study, would carry similar devices and benefit from similar satellite platform designs and orbits as the current gravity field missions. It is clear that there is a large potential to increase this synergy today, and opportunity for dual-purpose missions in the future. We therefore conclude by providing recommendations in terms of instrumentation and data handling for present and future missions.
Over the years, the linear accelerations measured by gravity missions have become an important source for thermospheric neutral density and horizontal wind data. By reducing the measured acceleration by models of thrust and radiation pressure, the aerodynamic acceleration remains, which is iteratively solved for the wind and density. A similar approach has now been taken to derive thermospheric wind from GOCE’s angular accelerations. In this case supplementary models were required for the magnetic torque (including attitude control) and gravity gradient torque. Because not all magnetic properties of GOCE were available to us, daily estimates were made of several magnetic dipoles on the satellite. On top of that a new iterative algorithm was developed that solves the residual force or torque for the neutral density and horizontal and vertical crosswind. The algorithm allows the use of any set of forces and torques. Horizontal winds derived from torques are found to agree up to a large extent with those derived from forces. Both the vertical force and the pitch torque reveal vertical wind signals near the poles. By combining wind data derived from forces with that derived from torques, a reliable vertical wind data set can be established. It is clear from the combined product that the vertical wind responds to increased geomagnetic activity. During geomagnetic storms, peak speeds are observed of up to 150 m/s upward and 100 m/s downward, a factor 5 smaller than the observed horizontal winds. Vertical wind structures are also spatially smaller than their horizontal counterparts, and therefore seem to have a more erratic nature. A correlation is occasionally found between horizontal and vertical wind peaks which is mostly lost when activity increases to high levels. Overall the newly derived data confirms the view that vertical wind structures are usually more local and short-lived than horizontal ones and might be a good indicator of small-scale wave activity.
...
Over the years, the linear accelerations measured by gravity missions have become an important source for thermospheric neutral density and horizontal wind data. By reducing the measured acceleration by models of thrust and radiation pressure, the aerodynamic acceleration remains, which is iteratively solved for the wind and density. A similar approach has now been taken to derive thermospheric wind from GOCE’s angular accelerations. In this case supplementary models were required for the magnetic torque (including attitude control) and gravity gradient torque. Because not all magnetic properties of GOCE were available to us, daily estimates were made of several magnetic dipoles on the satellite. On top of that a new iterative algorithm was developed that solves the residual force or torque for the neutral density and horizontal and vertical crosswind. The algorithm allows the use of any set of forces and torques. Horizontal winds derived from torques are found to agree up to a large extent with those derived from forces. Both the vertical force and the pitch torque reveal vertical wind signals near the poles. By combining wind data derived from forces with that derived from torques, a reliable vertical wind data set can be established. It is clear from the combined product that the vertical wind responds to increased geomagnetic activity. During geomagnetic storms, peak speeds are observed of up to 150 m/s upward and 100 m/s downward, a factor 5 smaller than the observed horizontal winds. Vertical wind structures are also spatially smaller than their horizontal counterparts, and therefore seem to have a more erratic nature. A correlation is occasionally found between horizontal and vertical wind peaks which is mostly lost when activity increases to high levels. Overall the newly derived data confirms the view that vertical wind structures are usually more local and short-lived than horizontal ones and might be a good indicator of small-scale wave activity.
Poster
(2018)
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Gunther March, Tim Visser, Eelco Doornbos, Eli Iorfida, Jose van den IJssel, Pieter Visser
Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.
...
Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.
Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.
...
Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.
In recent studies thermospheric densities and cross-winds have been derived from linear acceleration measurements of the gradiometer on board the GOCE satellite. Our current work is aimed at analyzing also the angular accelerations, in order to improve the thermosphere density and wind data by allowing for the estimation of more unknown parameters. On this poster an overview is provided of the modeling efforts involved in isolating the aerodynamic torque. The intermediate result is a comparison of modeled and measured torques. Each box contains a plot of the torque from a specific source, compared to the measured torque, on October 16th, 2013. A short description of the model for each torque is also provided.
...
In recent studies thermospheric densities and cross-winds have been derived from linear acceleration measurements of the gradiometer on board the GOCE satellite. Our current work is aimed at analyzing also the angular accelerations, in order to improve the thermosphere density and wind data by allowing for the estimation of more unknown parameters. On this poster an overview is provided of the modeling efforts involved in isolating the aerodynamic torque. The intermediate result is a comparison of modeled and measured torques. Each box contains a plot of the torque from a specific source, compared to the measured torque, on October 16th, 2013. A short description of the model for each torque is also provided.
In recent research efforts the multivariate simplex spline has shown great promise in system identification applications. It has high approximation power, while its linearity in the parameters allows for computationally efficient estimation of the coefficients. In this paper the multivariate simplotope spline is derived from this spline, and compared to its simplex counterpart in a system identification setting. Contrary to the simplex spline, the simplotope spline allows the user to incorporate expert knowledge of the system in his models. Whereas in the first spline all variables are included in a complete polynomial, in the latter the user can split the variables in decoupled subsets. By fitting models to specifically designed test functions it is shown that this can indeed improve the approximation performance in terms of both the error metrics and the number of B-coefficients required. This comes at the price of a higher total degree, and therefore an increased sensitivity to Runge's phenomenon in case of poor data distribution. Finally an attempt is made to apply the proposed methods to a set of flight data of the DelFly II, a flapping wing micro aerial vehicle. It is found that the used data set is not suitable for global system identification, as the data in concentrated in low-dimensional clusters in the five-dimensional state space. Therefore it is advised that a more suitable data set is obtained to validate the simplotope spline in a system identification setting.
...
In recent research efforts the multivariate simplex spline has shown great promise in system identification applications. It has high approximation power, while its linearity in the parameters allows for computationally efficient estimation of the coefficients. In this paper the multivariate simplotope spline is derived from this spline, and compared to its simplex counterpart in a system identification setting. Contrary to the simplex spline, the simplotope spline allows the user to incorporate expert knowledge of the system in his models. Whereas in the first spline all variables are included in a complete polynomial, in the latter the user can split the variables in decoupled subsets. By fitting models to specifically designed test functions it is shown that this can indeed improve the approximation performance in terms of both the error metrics and the number of B-coefficients required. This comes at the price of a higher total degree, and therefore an increased sensitivity to Runge's phenomenon in case of poor data distribution. Finally an attempt is made to apply the proposed methods to a set of flight data of the DelFly II, a flapping wing micro aerial vehicle. It is found that the used data set is not suitable for global system identification, as the data in concentrated in low-dimensional clusters in the five-dimensional state space. Therefore it is advised that a more suitable data set is obtained to validate the simplotope spline in a system identification setting.
Towards the multivariate simplotope spline
Continuity conditions in a class of mixed
Smooth joins of simplex Bernstein-B'ezier polynomials have been studied extensively in the past. In this paper a new method is proposed to define continuity conditions for tensor-product Bernstein polynomials on a class of mixed grids that meets certain out-of-facet parallelism criteria. The conditions are derived by first defining a simplex around the simplotopic bases of the tensor-product polynomials. Then the continuity conditions in the multivariate simplex spline defined on the resulting simplices, are adapted to hold for the tensor-product polynomials. The two- and three-dimensional results agree with the results found in the literature. It is expected that the method can be employed in more general grids.
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Smooth joins of simplex Bernstein-B'ezier polynomials have been studied extensively in the past. In this paper a new method is proposed to define continuity conditions for tensor-product Bernstein polynomials on a class of mixed grids that meets certain out-of-facet parallelism criteria. The conditions are derived by first defining a simplex around the simplotopic bases of the tensor-product polynomials. Then the continuity conditions in the multivariate simplex spline defined on the resulting simplices, are adapted to hold for the tensor-product polynomials. The two- and three-dimensional results agree with the results found in the literature. It is expected that the method can be employed in more general grids.