Maximizing the Onboard Capability of the Spacecraft Attitude Control System Based on Optimal Use of Reaction Wheels

2.1 RW configuration and arrangement

One of the purposes of optimization of RWs configuration is seeking of the maximum ability of torque/momentum (T/M) generation in R3 as maximum attainable volume. The boundary of this attainable volume is the T/M envelope of the RWs configuration. The optimal configuration defines only the relative orientations between RWs in a designing frame (DF) in which the attainable volume was calculated [13, 14]. After achieving the optimal configuration of RWs in the DF and the cluster is shaped, the arrangement of the RWs cluster in the body frame (BF) can be obtained.

2.2 Practical design considerations

For a particular RW configuration, a boundary of maximum T/M in all directions can be described as the maximum T/M capacity of this array. This is the physical limitation of this system which is defined by the vector sum of the maximum capacities of the individual wheels T/M. The best RWs configuration should maximize this volume. Thus, it ideally forms the most possible spherical shape. In practical, the T/M envelopes are not exact spheres but polyhedrons which can be skewed depending on the arrangement of the wheels array. If the momentum along any two axes is coupled depending on initial conditions or accumulated together due to secular perturbations, the momentum build-up can exceed either about the two axes or about any other combined directions in their plane. Therefore, for design purposes, the desired momentum capacity in this plane is required to be the same in all directions. If the desired momentum for the third axis is almost the same, the near-spherical momentum requirement for designing a wheel configuration will be obtained. In contrast, if the requirement for the third direction is small or has cyclic nature, this leads to right circular cylinder momentum requirement for designing a wheel configuration, with cylinder axis along the third axis. If a different peak control torque about each of the three SC axes is required, the torque requirement forms a rectangular parallelepiped.

2.3 Geometry of the envelope

A vector of n components can represent the T/M of n-wheel system. This T/M occupies the space of n-D hypercube, where the side length along the ith axis is twice the magnitude of the maximum T/M of the wheel along that axis, $T_{w i m a x} / H_{w i m a x}$ . Mapping the n-D hypercube into the three-dimensional (3D) space, the T/M fills a polyhedron involving sharp edges and vertices. This signifies that more T/M is available in certain directions than others. This T/M envelope produces the physical acceleration/rate limit of the system about any given axis when related to its mass properties. The 3D polyhedron vertices, edges, and facets are mappings of the hypercube vertices, edges, and facets. These mappings are not 1:1, and some hypercube vertices, edges, and facets are mapped to the interior of the 3D polyhedron [10, 19]. Vertices are points where all wheels are saturated. Edges have all but one wheels saturated. The edge with unsaturated wheel i is parallel to the spin axis unit vector of this wheel i ($\widehat{\mathrm{w}}_{\mathrm{i}}$) and has the length $2 \mathrm{H}_{\mathrm{wmaxi}}$ . Facets have all but two wheels saturated. The facet with unsaturated wheels i and j is a parallelogram, with sides parallel to $\widehat{\mathrm{w}}_{\mathrm{i}}$ and $\widehat{\mathrm{w}}_{\mathrm{j}}$ .

Markley [10] shows the 3D polyhedrons for symmetric configurations of the three-, four-, six-wheel, and the asymmetrical five-wheel case resulted from deletion of one wheel from the six-wheel configuration. Koh [12] shows the asymmetrical five-wheel case resulting from adding of one wheel to the four-wheel configuration in order to increase the capacity in the dominant maneuver direction.

If the unit length in the momentum envelope drawing is the maximum angular momentum capability of a single wheel and in the torque envelope drawing is the maximum torque capability of a single wheel, thus both envelopes are representing identically and labeled as torque envelope, momentum envelope, or envelope. If T/M capabilities of the wheels are changed, the shape of the T/M envelope will change but the structure will not. In this paper, equal wheels T/M capabilities will be chosen.

For n-wheel configuration, the polyhedron has $4 n(n-1)$  not distinct vertices [19], the wheel spin axis unit vectors defined by $\left\{\widehat{w}_{1} \widehat{w}_{1} \dots \widehat{w}_{n}\right\}$  and the control effectiveness matrix $w=\left[A_{w}\right]$  is

$w=\left[\widehat{w}_{1} \widehat{w}_{2} \dots \widehat{w}_{n}\right]$   (1)

For all pairs $i$ and $j$ of wheels in turn and wheel $k$ other than $i$ and $j(k \neq i, k \neq j),$ the $3 \mathrm{D}$ polyhedron can be characterized numerically [ 19] by $2^{n-2}$ parallel planes with normal vectors

$\hat{n}_{i j}=\frac{\hat{w}_{i} \times \hat{w}_{j}}{\left|\hat{w}_{i} \times \hat{w}_{j}\right|}$   (2)

The net wheel torque in the direction of the normal to the facet is

$\mathrm{T}_{i j}=d_{i j} T_{w \max }$   (3)

where,

$d_{i j}=\hat{\mathbf{n}}_{i j} \cdot \sum_{k \neq i, j} \widehat{w}_{k} \operatorname{sign}\left(\widehat{w}_{k} \cdot \hat{n}_{i j}\right)$    (4)

$\mathrm{T}_{i j}$ is the minimum torque capability on the facet ij, and the minimum value of $\mathrm{T}_{i j}$ over all the facets is the minimum torque capability of the RWs configuration. The parameter $\mathrm{d}_{i j}$ can characterize the RWs configuration. It represents the minimum T/M capability on the facet ij per unit length (UL) for drawing the envelope ($T_{\text {wimax}} / H_{\text {wimax}}$), and the minimum value of $\mathrm{d}_{i j}$  over all the facets is the minimum T/M capability of the RWs configuration per unit length which is defined as the radius of the largest inscribed sphere [19]. This is the physical system limit that cannot be exceeded.

2.4 Control input torque vector

The virtual control input torque vector $=\left[u_{x} u_{y} u_{z}\right]^{T}$ , that is desired to be produced along the three SC-body-axes, can be explained as the total control effort that is required to be produced by the actuators. It relates to the actual RWs torque vector $\tau_{w}=\left[\tau_{w 1}, \tau_{w 1}, \ldots, \tau_{w n}\right]^{T}$ for n-wheel configuration by the physical geometrical configuration as follows:

$[u]=\left[\begin{array}{l}{u_{x}} \\ {u_{y}} \\ {u_{z}}\end{array}\right]=\left[A_{w}\right]\left[\begin{array}{c}{\tau_{w 1}} \\ {\tau_{w 2}} \\ {\vdots} \\ {\tau_{w_{n}}}\end{array}\right]=\left[A_{w}\right]\left[\tau_{w}\right]$    (5)

where, $\left[A_{w}\right]$ is the torque distribution matrix (control effectiveness matrix). At least three column vectors of $\left[A_{w}\right]$ must be linearly independent or $\left(\left[A_{w}\right]\left[A_{w}\right]^{T}\right)^{-1}$  must exist for independent three-axis control.

In this section, the choice of using two assemblies, four RWs each, will be discussed. The geometric analysis of this eight RWs cluster is performed and the enhancement of the dynamic capability based on optimal use of this configuration will be discussed. The choice of two assemblies not only increases the capability of the control system when operating together (eight wheels) during imaging mode, but also gives the option of using one assembly only (four wheels) during no imaging modes instead of the total number of wheels. That saves the wheels power consumption at zero speed and torque. Additionally, smaller wheels may be more capable than less large wheels because the envelope would have more flat facets and less physical limitation losses. Thus for different RWs configurations of similar envelopes volumes, the large number RWs configuration allows a larger inscribed sphere as described [19]. To illustrate the previous paragraph, a comparison between two different clusters, eight RWs each, is performed, considering all RWs are in symmetry. Both clusters are consisting of two assemblies (four wheels each). The difference between the two clusters is the relative arrangement of assemblies inside a cluster. In the first cluster, shown in Figure 3a, both assemblies have the same arrangement where the spin axis of the first wheel in the second assembly coincides with the spin axis of the first wheel in the first assembly. Thus, the projections of the spin axes of the wheels on the X-Z plane inside this cluster are allocated twice each $90^{\circ} .$ While in the second cluster, shown in Figure $3 b,$ the projections of the spin axes of the first wheels in each assembly on the $\mathrm{X}-\mathrm{Z}$ plane are shifted $45^{\circ}$ from each other. Thus, the projections of the spin axes of all wheels are set radially symmetrically along the X-Z plane with configuration angle $\alpha_{w}=45^{\circ}$ for symmetry between the roll and pitch axes (uniformly distributed in azimuth about the Y-axis). Another skew configuration angle $\beta_{w}$ (cant angle) is the elevation angle from the $\mathrm{XZ}$ plane. The spin axes of all wheels are equally canted toward the Y-axis by an angle $\beta_{w}$ measured from the roll and pitch plane for both clusters. As the torque requirements about the $\mathrm{X}-$ and $\mathrm{Z}$ -axis increases, the optimum cant angle $\beta_{w}$ decreases.

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Figure 3. Eight wheels geometrical configurations

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Figure 4. 8-wheel configuration torque envelope, first cluster, 3D, XY, YZ, XZ Views

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Figure 5. 8-wheel configuration torque envelope, second cluster, 3D, XY, YZ, XZ Views

Figure 4 shows the torque envelope corresponding to the first cluster. Notice that the same envelope can be obtained from using four RWs with double maximum torque. Figure 5 shows the torque envelope corresponding to the second cluster. Assuming all wheels have equal T/M given by $T_{w \max } / H_{w \max }$ the dimensions of the envelopes are normalized and scaled symmetrically with the size of $T_{w \max } / H_{w \max } / H_{w \max } .$ In Figure $4,$ a) shows the total $\mathrm{T} / \mathrm{M}$ envelope; $\mathrm{b}$ ) shows the envelope in case of one wheel off, $\mathrm{c}$ ) shows the total envelope inscribed sphere, d) shows the inscribed sphere in case of one wheel off. In Figure $5 \mathrm{a}$ shows the total $\mathrm{T} / \mathrm{M}$ envelope, $\mathrm{b}$ ) shows the total envelope inscribed sphere, $\mathrm{c}$ ) shows the inscribed sphere in case of one $\alpha_{B F}=45[d e g]$ wheel off, d) shows the inscribed sphere in case of one $\alpha_{B F}=0,90[\operatorname{deg}]$ wheel off. All figures showed in $3 \mathrm{D},$ pitch-yaw $(\mathrm{XY}),$ yaw-roll $(\mathrm{YZ}),$ and pitch-roll $(\mathrm{XZ})$ views.

Table 2 summarizes several notes from Figure 4 and 5. The first cluster has maximum torque/UL value in X and Z axes of 4.6188 which is greater than 3.9424 of the second cluster. This increased value is due to the characteristic of $\alpha_{B F}=45[\operatorname{deg}]$ arrangement as described in Iable $1 .$ However, the secondcluster has 20.7 % more envelope volume, 10 % more inscribed sphere radius/UL in the case of all-wheel working, and 8.5 % more inscribed sphere radius/UL in case of one wheel off than the first cluster. That is enough to select the second cluster in order to increase the SC agility performance. The outcome envelope volume and inscribed sphere radius are greater than coincident assemblies. Comparing between the inscribed spheres in normal operating case for both clusters shown in Figure 4c, and 5b, it is clear that the inscribed sphere in 1st cluster is tangent to all center points of all facets while in 2nd cluster still there are several facets not reached yet by the inscribed sphere. Wheel failure destroys the symmetry of the shape and significantly reduces the envelope. The reduction in available torque in the direction of minimum capability are 27 %, 28 % of normal values for 1st, and 2nd clusters respectively.

Table 2. 8-wheel pyramid configuration

4.1 Control input torque vector for eight RWs configuration

This vector is limited by the available space of the torque envelope of the selected cluster and related to the actual RWs torque vector $\tau_{w}$ by the physical geometrical configuration illustrated in Figure 3 follows:

$[u]=\left[\begin{array}{c}{u_{x}} \\ {u_{y}} \\ {u_{z}}\end{array}\right]=\left[A_{w}\right]\left[\begin{array}{c}{\tau_{w 1}} \\ {\tau_{w 2}} \\ {\tau_{w_{3}}} \\ {\tau_{w 4}} \\ {\tau_{w_{5}}} \\ {\tau_{w_{6}}} \\ {\tau_{w_{8}}}\end{array}\right]=\left[A_{w}\right]\left[\tau_{w}\right]$   (8)

where, $\left[A_{w}\right]$ is the torque distribution matrix defined as

$\left[A_{w}\right]=\left[\begin{array}{cccc}{-c \beta c \alpha} & {-c \beta c \alpha} & {c \beta c \alpha} & {-c \beta c \alpha} & {-c \beta} & {0} & {c \beta} & {0}\\ {-c \alpha} & {c \alpha} & {-c \alpha} & {-c \alpha} & {-c \alpha} & {-c \alpha} & {-c \alpha} & {c \alpha} \\ {c \beta c \alpha} & {-c \beta c \alpha} & {-c \beta c \alpha} & {-c \beta c \alpha} & {0} & {c \beta} & {0} & {c \beta}\end{array}\right]$  (9)

Ignoring the actuator saturation and dynamics which is much faster than the remaining SC dynamics, the relation between the actual RWs torque vector $\tau_{w}$ produced from RWs and the commanded RWs torque vector $\tau_{w c}$ (the amplitude vector of allocated torque of each RW) can be approximated by the steady state relationship $\tau_{w}=\tau_{w c} .$ since the magnitudes of the available T/M may vary in different directions, and $\mathrm{SC}$ is not inertially symmetric in general, the magnitudes of the available body rate and acceleration also vary along the rotational directions. The optimality of agility performance is defined as the ability to achieve largest magnitudes of body rate and acceleration vectors along the worst direction (along which the vectors have minimum magnitude) [15].

4.2 Wheels configuration for optimal agility performance

For the shifted assemblies cluster, the cant angle $\beta_{w}$ can be set to the value which gives the same maximum torque along the SC body axes, $U_{i},(i=x, y, z)$,which are calculated as follows

$\begin{aligned} U_{x}=& \tau_{w \max } \cos \beta_{w}\left(4 \cos \alpha_{w}+2\right) \\ & U_{y}=8 \tau_{w_{-} \max } \sin \beta_{w} \\ U_{z}=& \tau_{wmax}\cos \beta_{w}\left(4 \sin \alpha_{w}+2\right) \end{aligned}$  (10)

for,

$U_{x}=U_{y}=U_{z}$   (11)

where, $\tau_{wmax}$ is the maximum RW torque. The optimalconfiguration angle $\beta_{w}$ can be obtained from equations ( 10) and ( 11) as

$\tan \beta_{w}=\frac{4 \cos \alpha_{w}+2}{8}$   (12)

So, the optimal configuration angle $\beta_{w}$ is set to $\beta_{w}=31.1^{\circ} .$ This choice is made without consideration of the inertia property of the SC.

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Figure 6. 8-wheel configuration torque envelope

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Figure 7. 8-wheel pyramid configuration relative to the corresponding inertia ellipsoid

Another choice of configuration angles can be made with consideration of the SC inertia [15]. It is based on calculating the optimal value of the angle $\beta_{w}$ by equating the body accelerations in 3-axes as follows

$\begin{aligned} a_{x_{-} \max }=\frac{U_{x}}{J_{x x}} &=\frac{\tau_{w_{-} \max } \cos \beta_{w}\left(4 \cos \alpha_{w}+2\right)}{J_{x x}} \\ a_{y_{-} \max } &=\frac{U_{y}}{J_{y y}}=\frac{8 \tau_{w_{-} \max } \sin \beta_{w}}{J_{y y}} \\ a_{z_{-} \max } &=\frac{U_{z}}{J_{z z}}=\frac{\tau_{w_{-} \max } \cos \beta_{w}\left(4 \sin \alpha_{w}+2\right)}{J_{z z}} \end{aligned}$   (13)

for,

$a_{x_{-} \max }=a_{y_{-} \max }=a_{z_{-} \max }$   (14)

where, $J_{x x}=780, J_{y y}=450,$ and $J_{z z}=780\left[K g \cdot m^{2}\right]$ are the SC inertia about principal axes, while the products of inertia are negligible. The optimal configuration angle $\beta_{w}$ can be obtained from equations ( 13) and ( 14) as

$\tan \beta_{w}=\frac{\left(4 \cos \alpha_{w}+2\right) I_{y y}}{8 J_{x x}}$    (15)

So, the optimal configuration angle $\beta_{w}$ is set to $\beta_{w} \approx 20^{\circ} .$ In the practical SC design, the configuration angles choice is a compromise among expected momentum accumulation, required torque, slewing requirements, and convenience of

fabrication. In Figure $6 \mathrm{a}$ shows the 8 -wheel pyramid configuration torque envelope for optimal configuration angles $\alpha_{w}=45, \beta_{w}=31.1[d e g],$ while Figure 66 shows the $\alpha_{w}=45, \beta_{w}=20[\operatorname{deg}]$ configuration. Figure $7 \mathrm{a},$ and $\mathrm{b}$ show the same configurations relative to the corresponding inertia ellipsoid in $3 D,$ Pitch-Yaw, Yaw-Roll, Pitch-Roll views Figure 7 illustrates that the envelope in the $\alpha_{w}=45, \beta_{w}=20[\text { deg}]$ configuration is more consistent to the SC inertia property than the other configuration. This is an indication of the appropriate usage of the available T/M capacity to achieve largest magnitudes of body rate and acceleration according to optimal agility performance definition.

4.3 Calculating maximum capability of RWs system

For the 8 -wheel pyramid configuration cluster of $\alpha_{w}=45, \beta_{w}=20[\operatorname{deg}]$ which is arranged as shown in Figure $3,$ the actual RWs torque vector $\tau_{w}$ consists of $i^{t h}$ RWs delivered torques assuming that.

$\left|\tau_{w i}\right| \leq \tau_{w i_{max} } \quad i=1: 8$    (16)

where, $\tau_{w i_{max }}=0.2[N m]$ is the maximum RW torque. Themaximum RWs torque along the SC body axes $U=\left[U_{x}, U_{x}, U_{x}\right]^{T}$ is calculated according to equations (10) and equal $(0.91 ; 0.55 ; 0.91)[N m]$. The maximum angular moment of individual RW is $H_{w \max }=18[\mathrm{N} . \mathrm{m} . \mathrm{s}] .$ The maximum angular momentum along the SC body axes $\mathrm{H}_{\mathrm{max}}$ is defined similar to equations (10) and equal  $(82 ; 50 ; 82)[\mathrm{Nms}]$. The maximum values of the absolute magnitudes of $\mathrm{SC}$ acceleration in 3 -axes are $a_{x_{ max}}=a_{y_{max }}=a_{z_{max}} \approx 0.067\left[\operatorname{deg} / s^{2}\right] .$ The maximum value of the absolute magnitudes of $\mathrm{SC}$ body rates in 3 -axes are $\omega_{x_{-} \max }=\omega_{y_{-} \max }=\omega_{z_{-} \max }=0.0059 H_{w \max } \approx 6[\operatorname{deg} /s]$. It should be noted that the maximum torque, momentum, acceleration or slew rates in 3-axes cannot be obtained simultaneously; instead, each of them is obtained when the total momentum is aligned along the corresponding body axis. Table 3 illustrates the results of the comparison between wheels configuration with and without SC inertia consideration according to the definition of optimality of agility performance. Selecting the wheels’ configuration by equating the body rates or accelerations in 3-axes can achieve, in this case study, 9.85 % increase in SC acceleration vectors along the worst direction as shown by gray color in Figure 8.

Table 3. Wheels configuration with/without consideration of the SC inertia

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Figure 8. Achieved acceleration according to maximum agility definition

4.4 Updating of system capability

Maximum system capability should be updated onboard according to RWs in use. Table 4 shows the maximum capability of RWs system in case of octahedron pyramid configuration cluster of shifted assemblies with consideration of the SC inertia $\alpha_{w}=45, \beta_{w}=20[\operatorname{deg}] .$ The eight wheels are assumed identical, maximum RW torque is $\tau_{w i \max }=0.2[N . m],$ inclined to the pitch-roll plane by same angle $\beta_{w}=20[d e g]$ and equally spaced around yaw axis by an angle $\alpha_{w}=45[\operatorname{deg}] .$ For this designed case with eight RW in use, the maximum values of the absolute magnitudes of SC acceleration and body rates are equal in 3 -axes and equal $0.067\left[\operatorname{deg} / s^{2}\right]$ and $6.1[\operatorname{deg} / s]$, respectively. By getting one wheel off or more, the equality of maximum values of $\mathrm{SC}$ acceleration in 3 -axes cannot be maintained. For one $\mathrm{RW}$ off from the first complex, equal reductions in maximum values of SC acceleration in pitch and roll directions are obtained as expected from angle $\alpha_{w}=45[\operatorname{deg}]$ and the maximum values of the absolute magnitudes of $\mathrm{SC}$ acceleration are reduced to $(0.057 ; 0.060 ; 0.057)\left[\operatorname{deg} / s^{2}\right] .$ For one $\mathrm{RW}$ off from the second complex, the reduction in maximum values of $\mathrm{SC}$ acceleration is in pitch or roll direction according to fault $\mathrm{RW}$ configuration. The maximum values of the absolute magnitudes of SC acceleration are reduced to $(0.053 ; 0.060 ;0.067)\left[\operatorname{deg} / s^{2}\right]$ in case of $\mathrm{RW} 5$ or RW7 fault and $(0.067 ;0.060 ; 0.053)\left[\operatorname{deg} / s^{2}\right]$ in case of RW6 or RW8 fault.

The used way to submit the attitude controller by the maximum values of the absolute magnitudes of SC acceleration in 3-axes is using predefined matrix. The elements of this predefined matrix are the worst case values corresponding to the number of wheels in use. For one RW off, the worst-case values are (0.053; 0.060; 0.053) $\left[\operatorname{deg} / s^{2}\right]$. The attitude controller is submitted with the allowable values of SC acceleration in 3-axes. These values can be calculated exactly depending not only on the number of wheels in use, but also on its configurations. When a fault occurs and one or more RWs are turned off, the corresponding columns in torque distribution matrix are nulled and the allowable acceleration is calculated according to the updated torque distribution matrix.

Table 4. Maximum capability of RWs system according to number of RWs in use

For the case of one RW fault, as illustrated in Table 4, the maximum values of the absolute magnitudes of SC acceleration are reduced to one of the following values (0.057; 0.060; 0.057), (0.053;0.060;0.067) or (0.067;0.060;0.053) according to fault RW instead of reduced to the worst case values (0.053;0.060;0.053) corresponding to the number of wheels in use. Using this dynamic way to calculate the maximum values of the absolute magnitudes of SC acceleration in 3-axes, up to 26.23 % increase in SC acceleration in the roll or pitch channels can be obtained in this case study for one RW off whereas up to 71.7 % increase in SC acceleration in roll or pitch channels for two RW off in comparison of the traditional way. An algorism can be used to implement this dynamic way for calculating the maximum values of the absolute magnitudes of SC acceleration in 3-axes in order to increase the used acceleration.

4.5 Forced drop of system capability

Traditionally, maximum capability of RWs system is calculated or selected based on the worst case values as shown in Table 4. The maximum capability according to the number of RWs in use for the cases of 5, 4, and 3 RWs was calculated. Results are summarized in Table 5, that shows the forced capability reduction in case of three RW fault (5 working RW) to the capability of just 4 RW working. In the RW octahedron cluster pyramid configuration, the forced drop lagged till third RW fault. While in other configurations, this drop may appear from the first RW fault.

Table 5. Brief results of maximum capability for 5,4, and 3 RWs

4.6 Combined enhancement of system capability

The dynamic capability can be enhanced due to the combination of the capability enhancement based on selecting wheels configuration with consideration of the SC inertia $\left(\alpha_{w}=45[\operatorname{deg}], \beta_{w}=20[\operatorname{deg}]\right)$ and capability update based on torque distribution matrix update with RW fault.  The obtained enhancement can be analyzed as follows;

For no RW fault, the capability enhancement is only based on selecting wheels configuration with consideration of the SC inertia $\left(\alpha_{w}=45[\operatorname{deg}], \beta_{w}=20[\operatorname{deg}]\right)$. The enhancement in this case study equals 9.85 %.

For one RW fault, the combined capability enhancement is illustrated in Table 6. The enhancement in this case study can reach to 38.51 % in pitch or roll channels.

Table 6. Combined capability enhancement