diff --git a/test-bench-nano-hexapod.html b/test-bench-nano-hexapod.html index ccff1c2..35adafe 100644 --- a/test-bench-nano-hexapod.html +++ b/test-bench-nano-hexapod.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Nano-Hexapod - Test Bench @@ -39,54 +39,54 @@

Table of Contents

@@ -95,12 +95,19 @@

-In this document, the dynamics of the nano-hexapod shown in Figure 1 is identified. +This document is dedicated to the experimental study of the nano-hexapod shown in Figure 1.

-
+ +
+

IMG_20210608_152917.jpg +

+

Figure 1: Nano-Hexapod

+
+ +

-Here are the documentation of the equipment used for this test bench: +Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure 2):

-
-

IMG_20210608_152917.jpg -

-

Figure 1: Nano-Hexapod

-
- - -
+

IMG_20210608_154722.jpg

Figure 2: Nano-Hexapod and the control electronics

+

+In Figure 3 is shown a block diagram of the experimental setup. +When possible, the notations are consistent with this diagram and summarized in Table 1. +

-
+ +

nano_hexapod_signals.png

Figure 3: Block diagram of the system with named signals

- +
@@ -250,25 +255,50 @@ Here are the documentation of the equipment used for this test bench:
Table 1: List of signals
-
-

1 Encoders fixed to the Struts

+

+This document is divided in the following sections: +

+
    +
  • Section 1: the encoders are fixed to the struts
    • +
  • Section 2: the encoders are fixed to the plates
    • +
+ +
+

1 Encoders fixed to the Struts

+

+ +

-
-

1.1 Introduction

+ +
+

1.1 Introduction

In this section, the encoders are fixed to the struts.

+ +

+It is divided in the following sections: +

+
    +
  • Section 1.2: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified.
    • +
  • Section 1.3: the obtained FRF matrix is compared with the dynamics of the simscape model
    • +
  • Section 1.4: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated.
    • +
  • Section 1.5: a modal analysis of the nano-hexapod is performed
    • +
-
-

1.2 Identification of the dynamics

+
+

1.2 Identification of the dynamics

+

+ +

-
-

1.2.1 Load Data

+
+

1.2.1 Load Data

%% Load Identification Data
@@ -283,8 +313,8 @@ meas_data_lf = {};
-
-

1.2.2 Spectral Analysis - Setup

+
+

1.2.2 Spectral Analysis - Setup

%% Setup useful variables
@@ -307,11 +337,11 @@ i_hf = f > 250; % Poi
-
-

1.2.3 DVF Plant

+
+

1.2.3 DVF Plant

-First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 4). +First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 4).

@@ -327,14 +357,14 @@ coh_dvf_hf = zeros(length(f), 6, 6);
-
+

enc_struts_dvf_coh.png

Figure 4: Obtained coherence for the DVF plant

-Then the 6x6 transfer function matrix is estimated (Figure 5). +Then the 6x6 transfer function matrix is estimated (Figure 5). 

%% DVF Plant (transfer function from u to dLm)
@@ -349,7 +379,7 @@ G_dvf_hf = zeros(length(f), 6, 6);
-
+

enc_struts_dvf_frf.png

Figure 5: Measured FRF for the DVF plant

@@ -358,11 +388,11 @@ G_dvf_hf = zeros(length(f), 6, 6);
-
-

1.2.4 IFF Plant

+
+

1.2.4 IFF Plant

-First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 6). +First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 6).

@@ -379,14 +409,14 @@ coh_iff_hf = zeros(length(f), 6, 6);
-
+

enc_struts_iff_coh.png

Figure 6: Obtained coherence for the IFF plant

-Then the 6x6 transfer function matrix is estimated (Figure 7). +Then the 6x6 transfer function matrix is estimated (Figure 7). 

%% IFF Plant
@@ -401,7 +431,7 @@ G_iff_hf = zeros(length(f), 6, 6);
-
+

enc_struts_iff_frf.png

Figure 7: Measured FRF for the IFF plant

@@ -410,16 +440,18 @@ G_iff_hf = zeros(length(f), 6, 6);
-
-

1.3 Comparison with the Simscape Model

+
+

1.3 Comparison with the Simscape Model

+ +

+

In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.

- -
-

1.3.1 Dynamics from Actuator to Force Sensors

+
+

1.3.1 Dynamics from Actuator to Force Sensors

%% Initialize Nano-Hexapod
@@ -441,14 +473,14 @@ Giff = exp(-s*Ts)
 
 
-

+

 
enc_struts_iff_comp_simscape.png
 

 
Figure 8: Diagonal elements of the IFF Plant

 

 
 
-

+

 
enc_struts_iff_comp_offdiag_simscape.png
 

 
Figure 9: Off diagonal elements of the IFF Plant

@@ -456,8 +488,8 @@ Giff = exp(-s*Ts)
 

 
-

-
1.3.2 Dynamics from Actuator to Encoder

+

+
1.3.2 Dynamics from Actuator to Encoder

 

 

 
%% Initialization of the Nano-Hexapod
@@ -479,14 +511,14 @@ Gdvf = exp(-s*Ts)
 
 
-

+

 
enc_struts_dvf_comp_simscape.png
 

 
Figure 10: Diagonal elements of the DVF Plant

 

 
 
-

+

 
enc_struts_dvf_comp_offdiag_simscape.png
 

 
Figure 11: Off diagonal elements of the DVF Plant

@@ -495,12 +527,15 @@ Gdvf = exp(-s*Ts)
 

 
-

-
1.4 Integral Force Feedback

+

+
1.4 Integral Force Feedback

 

+

+
+

 

-

-
1.4.1 Root Locus and Decentralized Loop gain

+

+
1.4.1 Root Locus and Decentralized Loop gain

 

 

 
%% IFF Controller
@@ -512,7 +547,7 @@ Kiff_g1 = (1/(s + 2<
 

 
 
-

+

 
enc_struts_iff_root_locus.png
 

 
Figure 12: Root Locus for the IFF control strategy

@@ -528,7 +563,7 @@ Kiff = g*Kiff_g1;
 

 
 
-

+

 
enc_struts_iff_opt_loop_gain.png
 

 
Figure 13: Bode plot of the “decentralized loop gain” \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)

@@ -536,8 +571,8 @@ Kiff = g*Kiff_g1;
 

 

 
-

-
1.4.2 Multiple Gains - Simulation

+

+
1.4.2 Multiple Gains - Simulation

 

 

 
%% Tested IFF gains
@@ -573,7 +608,7 @@ io(io_i) = linio([mdl, '/D'],  1, 
+

 
enc_struts_iff_gains_effect_dvf_plant.png
 

 
Figure 14: Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)

@@ -581,16 +616,16 @@ io(io_i) = linio([mdl, '/D'],  1, 
-
1.4.3 Experimental Results - Gains

+

+
1.4.3 Experimental Results - Gains

 

 

 Let’s look at the damping introduced by IFF as a function of the IFF gain and compare that with the results obtained using the Simscape model.
 

 

 
-

-
1.4.3.1 Load Data

+

+
1.4.3.1 Load Data

 

 

 
%% Load Identification Data
@@ -604,8 +639,8 @@ meas_iff_gains = {};
 

 

 
-

-
1.4.3.2 Spectral Analysis - Setup

+

+
1.4.3.2 Spectral Analysis - Setup

 

 

 
%% Setup useful variables
@@ -625,8 +660,8 @@ win = hanning(ceil(1*Fs));
 

 

 
-

-
1.4.3.3 DVF Plant

+

+
1.4.3.3 DVF Plant

 

 

 
%% DVF Plant (transfer function from u to dLm)
@@ -639,20 +674,20 @@ G_iff_gains = {};
 

 
 
-

+

 
comp_iff_gains_dvf_plant.png
 

 
Figure 15: Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain

 

 
 
-

+

 
comp_iff_gains_dvf_plant_zoom.png
 

 
Figure 16: Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain (Zoom)

 

 
-

+

 

 The IFF control strategy is very effective for the damping of the suspension modes.
 It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA).
@@ -667,29 +702,40 @@ Also, the experimental results and the models obtained from the Simscape model a
 

 

 
-

-
1.4.3.4 Experimental Results - Comparison of the un-damped and fully damped system

+

+
1.4.3.4 Experimental Results - Comparison of the un-damped and fully damped system

 

 
-

+

 
comp_undamped_opt_iff_gain_diagonal.png
 

 
Figure 17: Comparison of the diagonal elements of the tranfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) without active damping and with optimal IFF gain

+

+
+

+

+A series of modes at around 205Hz are also damped.
+

+
+

+Are these damped modes at 205Hz additional “suspension” modes or flexible modes of the struts?
+

+
 

 

 

 

 
-

-
1.4.4 Experimental Results - Damped Plant with Optimal gain

+

+
1.4.4 Experimental Results - Damped Plant with Optimal gain

 

 

 Let’s now look at the \(6 \times 6\) damped plant with the optimal gain \(g = 400\).
 

 

 
-

-
1.4.4.1 Load Data

+

+
1.4.4.1 Load Data

 

 

 
%% Load Identification Data
@@ -703,8 +749,8 @@ meas_iff_struts = {};
 

 

 
-

-
1.4.4.2 Spectral Analysis - Setup

+

+
1.4.4.2 Spectral Analysis - Setup

 

 

 
%% Setup useful variables
@@ -724,8 +770,8 @@ win = hanning(ceil(1*Fs));
 

 

 
-

-
1.4.4.3 DVF Plant

+

+
1.4.4.3 DVF Plant

 

 

 
%% DVF Plant (transfer function from u to dLm)
@@ -738,23 +784,23 @@ G_iff_opt = {};
 

 
 
-

+

 
damped_iff_plant_comp_diagonal.png
 

 
Figure 18: Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

 

 
 
-

+

 
damped_iff_plant_comp_off_diagonal.png
 

 
Figure 19: Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)

 

 
-

+

 

-With the IFF control strategy applied and the optimal gain used, the suspension modes are very well dapmed.
-Remains the undamped flexible modes of the APA, and the modes of the plates.
+With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped.
+Remains the undamped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz).
 

 
 

@@ -767,34 +813,36 @@ The Simscape model and the experimental results are in very good agreement.
 

 

 
-

-
1.5 Modal Analysis

+

+
1.5 Modal Analysis

 

 

-Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure 22.
+
+

+

+Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure 22.
 

 
 
-

+

 
accelerometers_nano_hexapod.jpg
 

 
Figure 20: Location of the accelerometers on top of the nano-hexapod

 

 
 

-The top platform is then excited using an instrumented hammer as shown in Figure 21.
+The top platform is then excited using an instrumented hammer as shown in Figure 21.
 

 
 
-

+

 
hammer_excitation_compliance_meas.jpg
 

 
Figure 21: Example of an excitation using an instrumented hammer

 

 

-
-

-
1.5.1 Effectiveness of the IFF Strategy - Compliance

+

+
1.5.1 Effectiveness of the IFF Strategy - Compliance

 

 

 In this section, we wish to estimated the effectiveness of the IFF strategy concerning the compliance.
@@ -828,28 +876,32 @@ d_frf_iff = 10/5*(fr
 

 
 

-The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure 22.
+The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure 22.
 

 
 
-

+

 
compliance_vertical_comp_iff.png
 

 
Figure 22: Measured vertical compliance with and without IFF

 

 
-

+

 

-From Figure 22, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapode.
+From Figure 22, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapode.
 It also has the effect of degrading (slightly) the vertical compliance at low frequency.
 

 
+

+It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts.
+

+
 

 

 

 
-

-
1.5.2 Comparison with the Simscape Model

+

+
1.5.2 Comparison with the Simscape Model

 

 

 Let’s now compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model.
@@ -857,11 +909,11 @@ Let’s now compare the measured vertical compliance with the vertical compl
 
 

 The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model.
-The comparison is done in Figure 23.
-Again, the model is quire accurate!
+The comparison is done in Figure 23.
+Again, the model is quite accurate!
 

 
-

+

 
compliance_vertical_comp_model_iff.png
 

 
Figure 23: Measured vertical compliance with and without IFF

@@ -869,40 +921,40 @@ Again, the model is quire accurate!
 

 

 
-

-
1.5.3 Obtained Mode Shapes

+

+
1.5.3 Obtained Mode Shapes

 

 

 Then, several excitation are performed using the instrumented Hammer and the mode shapes are extracted.
 

 
 

-We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure 24.
+We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure 24.
 

 
 
-

+

 
mode_shapes_annotated.gif
 

 
Figure 24: Measured mode shapes for the first six modes

 

 
 

-Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure 24).
+Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure 24).
 

 
 
-

+

 
ModeShapeFlex1_crop.gif
 

 
Figure 25: First flexible mode at 692Hz

 

 
 

-The obtained modes are summarized in Table 2.
+The obtained modes are summarized in Table 2.
 

 
-
+
@@ -914,8 +966,8 @@ The obtained modes are summarized in Table 2.
 
-
-
+
+
@@ -923,13 +975,13 @@ The obtained modes are summarized in Table 2.
 
-
+
-
+
@@ -941,13 +993,13 @@ The obtained modes are summarized in Table 2.
 
-
+
-
+
@@ -968,13 +1020,18 @@ The obtained modes are summarized in Table 2.
 
 
-

-
2 Encoders fixed to the plates

+

+
2 Encoders fixed to the plates

+

+

+
+

+

 

 

 
Author: Dehaeze Thomas

-
Created: 2021-06-14 lun. 17:24

+
Created: 2021-06-14 lun. 18:07

 

diff --git a/test-bench-nano-hexapod.org b/test-bench-nano-hexapod.org
index 23d4697..36a5366 100644
--- a/test-bench-nano-hexapod.org
+++ b/test-bench-nano-hexapod.org
@@ -49,10 +49,15 @@
 #+latex: \clearpage
 
 * Introduction                                                        :ignore:
-In this document, the dynamics of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]] is identified.
+This document is dedicated to the experimental study of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]].
+
+#+name: fig:picture_bench_granite_nano_hexapod
+#+caption: Nano-Hexapod
+#+attr_latex: :width \linewidth
+[[file:figs/IMG_20210608_152917.jpg]]
 
 #+begin_note
-Here are the documentation of the equipment used for this test bench:
+Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure [[fig:picture_bench_granite_overview]]):
 - Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]]
 - Amplified Piezoelectric Actuator: Cedrat [[file:doc/APA300ML.pdf][APA300ML]]
 - DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
@@ -60,16 +65,14 @@ Here are the documentation of the equipment used for this test bench:
 - Interferometers: Attocube
 #+end_note
 
-#+name: fig:picture_bench_granite_nano_hexapod
-#+caption: Nano-Hexapod
-#+attr_latex: :width \linewidth
-[[file:figs/IMG_20210608_152917.jpg]]
-
 #+name: fig:picture_bench_granite_overview
 #+caption: Nano-Hexapod and the control electronics
 #+attr_latex: :width \linewidth
 [[file:figs/IMG_20210608_154722.jpg]]
 
+In Figure [[fig:nano_hexapod_signals]] is shown a block diagram of the experimental setup.
+When possible, the notations are consistent with this diagram and summarized in Table [[tab:list_signals]].
+
 #+begin_src latex :file nano_hexapod_signals.pdf
 \definecolor{instrumentation}{rgb}{0, 0.447, 0.741}
 \definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098}
@@ -112,7 +115,6 @@ Here are the documentation of the equipment used for this test bench:
 #+name: fig:nano_hexapod_signals
 #+caption: Block diagram of the system with named signals
 #+attr_latex: :scale 1
-#+RESULTS:
 [[file:figs/nano_hexapod_signals.png]]
 
 #+name: tab:list_signals
@@ -136,10 +138,22 @@ Here are the documentation of the equipment used for this test bench:
 | Motion of the top platform         | =[m,rad]= | =dX=      | $d\bm{\mathcal{X}}$   | $d\mathcal{X}_i$     |
 | Metrology measured displacement    | =[m,rad]= | =dXm=     | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ |
 
+This document is divided in the following sections:
+- Section [[sec:encoders_struts]]: the encoders are fixed to the struts
+- Section [[sec:encoders_plates]]: the encoders are fixed to the plates
+
 * Encoders fixed to the Struts
+<>
+
 ** Introduction
 In this section, the encoders are fixed to the struts.
 
+It is divided in the following sections:
+- Section [[sec:enc_struts_plant_id]]: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified.
+- Section [[sec:enc_struts_comp_simscape]]: the obtained FRF matrix is compared with the dynamics of the simscape model
+- Section [[sec:enc_struts_iff]]: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated.
+- Section [[sec:enc_struts_modal_analysis]]: a modal analysis of the nano-hexapod is performed
+
 ** Matlab Init                                             :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <>
@@ -161,6 +175,7 @@ addpath('./src/');
 #+end_src
 
 ** Identification of the dynamics
+<>
 *** Load Data
 #+begin_src matlab
 %% Load Identification Data
@@ -614,6 +629,7 @@ exportFig('figs/enc_struts_iff_cart_frf.pdf', 'width', 'wide', 'height', 'tall')
 [[file:figs/enc_struts_iff_cart_frf.png]]
 
 ** Comparison with the Simscape Model
+<>
 *** Introduction                                                    :ignore:
 In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
 
@@ -893,6 +909,7 @@ exportFig('figs/enc_struts_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'hei
 [[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]]
 
 ** Integral Force Feedback
+<>
 *** Root Locus and Decentralized Loop gain
 #+begin_src matlab
 %% IFF Controller
@@ -1263,6 +1280,12 @@ exportFig('figs/comp_undamped_opt_iff_gain_diagonal.pdf', 'width', 'wide', 'heig
 #+RESULTS:
 [[file:figs/comp_undamped_opt_iff_gain_diagonal.png]]
 
+#+begin_question
+A series of modes at around 205Hz are also damped.
+
+Are these damped modes at 205Hz additional "suspension" modes or flexible modes of the struts?
+#+end_question
+
 *** Experimental Results - Damped Plant with Optimal gain
 **** Introduction                                                  :ignore:
 Let's now look at the $6 \times 6$ damped plant with the optimal gain $g = 400$.
@@ -1435,13 +1458,14 @@ exportFig('figs/damped_iff_plant_comp_off_diagonal.pdf', 'width', 'wide', 'heigh
 [[file:figs/damped_iff_plant_comp_off_diagonal.png]]
 
 #+begin_important
-With the IFF control strategy applied and the optimal gain used, the suspension modes are very well dapmed.
-Remains the undamped flexible modes of the APA, and the modes of the plates.
+With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped.
+Remains the undamped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz).
 
 The Simscape model and the experimental results are in very good agreement.
 #+end_important
 
 ** Modal Analysis
+<>
 *** Introduction                                                    :ignore:
 Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure [[fig:compliance_vertical_comp_iff]].
 
@@ -1503,6 +1527,8 @@ exportFig('figs/compliance_vertical_comp_iff.pdf', 'width', 'wide', 'height', 'n
 #+begin_important
 From Figure [[fig:compliance_vertical_comp_iff]], it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapode.
 It also has the effect of degrading (slightly) the vertical compliance at low frequency.
+
+It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts.
 #+end_important
 
 *** Comparison with the Simscape Model
@@ -1539,7 +1565,7 @@ G_compl_z_iff = linearize(mdl, io, 0.0, options);
 #+end_src
 
 The comparison is done in Figure [[fig:compliance_vertical_comp_model_iff]].
-Again, the model is quire accurate!
+Again, the model is quite accurate!
 #+begin_src matlab :exports none
 %% Comparison of the measured compliance and the one obtained from the model
 freqs = 2*logspace(1,3,1000);
@@ -1601,4 +1627,6 @@ The obtained modes are summarized in Table [[tab:description_modes]].
 |    7 |        692 | (flexible) Membrane mode of the top platform |
 
 * Encoders fixed to the plates
+<>
+
 ** Introduction                                                      :ignore:
diff --git a/test-bench-nano-hexapod.pdf b/test-bench-nano-hexapod.pdf
index 3a30fc5..a0367be 100644
Binary files a/test-bench-nano-hexapod.pdf and b/test-bench-nano-hexapod.pdf differ


 Table 2: Description of the identified modes
 
 

 
 
Mode NumberFrequency [Hz]ModeFreq. [Hz]
 Description
 

 

 1
 105Suspension Mode: ~Y-translationSuspension Mode: Y-translation
 

 
 

 2
 107Suspension Mode: ~X-translationSuspension Mode: X-translation
 

 
 

 4
 161Suspension Mode: ~Y-tiltSuspension Mode: Y-tilt
 

 
 

 5
 162Suspension Mode: ~X-tiltSuspension Mode: X-tilt