Develop Reference

Detect Forward/Backward Movement on a Phone

I am using the accelerometer data to detect whether the phone is moving forward or backward. Assuming that the phone is held vertically parallel to the person's body, can we detect whether the phone is moving forward or backward while the person is moving?
Known parameters are:
Linear acceleration along all the three axes.
Ideally, Acceleration values should turn out to be positive when moving backward and negative while moving forward. But observing the values, it does not seem so. Is considering only acceleration about 'z' axis wrong?
Referred Questions:
Detect device forward backward movement => How do we create a direction vector and what is the need to monitor x-axis too?

You cannot assume, that you only have movement on z axis (be it now negative or positive, depending on starting or stopping), as the person will necessarily also move the phone (a little) to the side (x negative and positive) and up or down (y positive and negative) with each step. But the acceleration in z negative or positive should be a little stronger with each step. I'd calculate the means of around 10 to 20 steps before being 'sure' - better combine it with gps data.

Measure sideways tilt of upright device

Some camera apps have a feature where they display a line on the screen which is always parallell to the horizon, no matter how the phone is tilted sideways. By "tilted sideways" I mean rotating the device around an axis that is perpendicular to the screen.
I have tried most of the "normal" rotation functions such as combining the ROTATION_VECTOR sensor with getRotationMatrix() and getOrientation() but none of the resulting axis seem to correspond to the one I'm looking for.
I also tried using only the accelerometer, normalizing all three axis and detecting how much gravity is in the X-axis. This works decently when the device is perfectly upright (i.e not tilted forward/backwards). But as soon as it's tilted forward or backward the measured sideways tilt gets increasingly inaccurate, since gravity is now acting on two axis at the same time.
Any ideas on how to achieve this kind of sideways rotation detection in a way that works even if the phone is tilted/pitched slightly forward/backward?

The result of getRotationMatrix converts from the phone's local coordinate system to the world coordinate system. Its columns are therefore the principal axes of the phone, with the first being the phone's X-axis (the +ve horizontal axis when holding the phone in portrait mode), and the second being the Y.
To obtain the horizon's direction on the phone's screen, the line of intersection between the horizontal plane (world space) and the phone's plane must be found. First find the coordinates of the world Z-axis (pointing to the sky) in the phone's local basis - i.e. transpose(R) * [0, 0, 1]; the XY coordinates of this, given by R[2][0], R[2][1], is the vertical direction in screen-space. The required horizon line direction is then R[2][1], -R[2][0].
When the phone is close to being horizontal, this vector becomes very small in magnitude - the horizon is no-longer well-defined. Simply stop updating the horizon line below some threshold.

Unity device movement

I'm trying to move a game object when i raise/lower (Shake) my phone but I don't know how get the device's movement. I already know about Input.acceleration but that just gives me the devices rotation and I want it's actual movement.
Is this possible and how would I go about doing it?

Accelerometer reads the sum of: changes in movement (acceleration) + constant force of gravity. You can use it for directional movement in two ways:
Detect changes of gravity angle - when the device is not moving (or is moving at a constant speed) and is parallel to the ground, the accelerometer will read Earth's gravity, ie: new Vector3(0,-9.81,0). When the device is tilted (so not parallel to the ground), the vector's length will still be 9.81, but it will be rotated a bit like. new Vector3(3.03, -9.32, 0) (this example is rotation in one axis by 56 degrees (pi/10)). Using this will yield this kind of controlls: https://www.youtube.com/watch?v=3EU3ip4k0uE
Detect peaks of acceleration. When the device is still or moving with constant speed, the length of acceleration vector will be equal 9.81, when it changes/starts movement, this number will change. You can detect these changes and interpret this as one time momentary movement (like pressing an arrow).
There are alternatives not using an accelometer, for example you can detect some printed marker with Vuforia: https://www.youtube.com/watch?v=8RnlBEU5tkI - interpret the relative position to the marker as and action in a simmilar fashion as you'd detect acceleration change in #1.

detect the force (and not the acceleration force) applied to the Smartphone

I'm trying to measure the force applied to the device along the x axis.
If you push the device to the left (so the phone moves to the right) at constant speed,
measuring the acceleration (using TYPE_LINEAR_ACCELERATION) along the x axis will return positive number at the beginning, but very fast the acceleration will be equal to zero (again, pushing at a constant speed).
As explained in the API Ad = - ∑Fs / Mass , so I thought to achieve the force by multiplying the acceleration sensor result (event.values[0]) with the mass,
but multiplying acceleration equals to zero won't help.
The reason I'm trying to measure the force, and not the acceleration is I want to detect a long (in matter of time) gesture of moving the phone in constant speed, contradiction to shake for example.
The goal is to move a ball according to the movement of the phone.
Pushing the phone from the user:
Will move the ball upwards from it current position:

I'm no physicist, but I don't think you can do that. If an object is moving at a constant speed, it has no net force acting on it. Force is defined by acceleration and mass.
However, you can tell if the phone is still moving. If you see a rise in acceleration followed by a drop to zero, it's still moving. When the phone stops moving, you'll get an acceleration in the opposite direction equal to the starting acceleration.
So, if you move your ball up when you see an acceleration in that direction, don't just move it up by the current value. Sum the accelerations you've seen so far(weighted by time between sensor readings?). That way when it drops to zero, it will stay put. When it starts to accelerate the other way (slowing down), it will subtract from the position until it's roughly centered again.
Of course, the accelerometer is prone to drift, etc, so it's hard to get this perfect.

Vertical movement sensor

I am working on an android app that requires the detection of vertical motion. When moving the tablet upward, the Gyroscope, Accelerometer, and Linear Acceleration sensors give a corresponding value indicating upward or downward motion.
The problem I have is that these sensors will also read an upward/downward motion when you tilt the tablet towards the user or away from the user. For example, the x value in the gyroscope represents the vertical plane. But when you tilt the device forwards, the x value will change.
When I make this motion, the same sensor that reads vertical motion reads a value for this.
The same goes for the rest of the sensors. I have tried to use orientation coupled with the gyro to make the conditional statement, if the pitch is not changing, but the x variable is going up/down, then we have vertical motion. The problem with this is that if the user moves it up but tilted slightly, it will no longer work. I also tried making it so if there is a change in tilt, then there is no vertical motion. But it iterates so quickly that there may be a change in tilt for 1/100 of a second, but for the next there isn't.
Is there any way I can read only vertical changes and not changes in the devices pitch?
Here is what I want to detect:
edit:
"Please come up with a mathematically sound definition of what you consider 'moving upwards.'"
This was my initial question, how can I write a function to define when the tablet is moving upwards or downwards? I consider a vertical translation moving upwards. Now how do I detect this? I simply do not know where to begin, thank you.

Ok, even though this question is fairly old, I see a lot of confusion in the present answer and comments, so in case anyone finds this, I intend to clear a few things up.
The Gyroscope
First of all, the gyroscope does not measure vertical motion as per your definition (a translatory motion). It measures rotation around each of the axes, which are defined as in the figure below. Thus having you tilt your device forwards and backwards indeed rotates it around the x axis and therefore you will see non-zero values in the x value of your gyroscope sensor.
the x value in the gyroscope represents the vertical plane.
I'm not sure what is meant by "the vertical plane", however the x value certainly does not represent the plane itself nor the orientation of the device within the plane.
The x value of the gyroscope sensor represents the current angular velocity of the device around the x axis (eg. the change in rotation).
But when you tilt the device forwards, the x value will change. When I make this motion, the same sensor that reads vertical motion reads a value for this.
Not quite sure what you're referring to here. "The same sensor that reads vertical motion" I assume is the gyroscope, but as previously said, it does not read vertical motion. It does exactly what it says on the tin.
The device coordinate system
This is more in response to user Ali's answer than the original question, but it remains relevant in either case.
The individual outputs of the linear acceleration sensor (or any other sensor for that matter) are expressed in the coordinate system of the device, as shown in the image above. This means if you rotate the device slightly, the outputs will no longer be parallel to any world axis they coincided with before. As such, you will either have to enforce that the device is in a particular orientation for your application, or take the new orientation into account.
The ROTATION_VECTOR sensor, combined with quaternion math or the getRotationMatrixFromVector() method, is one way to translate your measurements from device coordinates to world coordinates. There are other ways to achieve the same goal, but once achieved, the way you hold your device won't matter for measuring vertical motion.
In either case, the axis you're looking for is the y axis, not the z axis.
(If by any chance you meant "along device y axis" as "vertical", then just ignore all the orientation stuff and just use the linear acceleration sensor)
Noise
You mentioned some problems regarding noise and update rates in the question, so I'll just mention it here. The simplest and one of the more common ways to get nice, consistent data from something that varies very often is to use a low-pass filter. What type of filter is best depends on the application, but I find that a exponential moving average filter is viable in most cases.
Finishing thoughts
Note that if you take proper care of the orientation, your transformed linear acceleration output will be a good approximation of vertical motion (well, change in motion) without filtering any noise.
Also, if you want to measure vertical "motion", as in velocity, you need to integrate the accelerometer output. For various reasons, this doesn't really turn out too well in most cases, although it is less severe in the case of velocity rather than trying to measure position.

OK, I suspect it is only a partial answer.
If you want to detect vertical movement, you only need linear acceleration, the device orientation doesn't matter. See
iOS - How to tell if device is raised/dropped (CoreMotion)
or
how to calculate phone's movement in the vertical direction from rest?
For some reason you are concerned with the device orientation as well, and I have no idea why. I suspect that you want to detect something else. So please tell us more and then I will improve my answer.
UPDATE
I read the post on coremotion, and you mentioned that higher z lower x and y means vertical motion, can you elaborate?
I will write in pseudo code. You measured the (x, y, z) linear acceleration vector. Compute
rel_z = z/sqrt(x^2+y^2+z^2+1.0e-6)
If rel_z > 0.9 then the acceleration towards the z direction dominates (vertical motion). Note that the constant 0.9 is arbitrary and may require tweaking (should be a positive number less than 1). The 1.0e-6 is there to avoid accidental division by zero.
You may have to add another constraint that z is sufficiently large. I don't know your device, whether it measures gravity as 1 or 9.81. I assume it measures it as 1.
So all in all:
if (rel_z > 0.9 && abs(z) > 0.1) { // we have vertical movement
Again, the constant 0.1 is arbitrary and may require tweaking. It should be positive.
UPDATE 2
I do not want this because rotating it towards me is not moving it upwards
It is moving upwards: The center of mass is moving upwards. My code has the correct behavior.
Please come up with a mathematically sound definition of what you consider "moving upwards."