Freescale Semiconductor
Application Note
Document Number: AN4446
Rev. 0, 08/2012
Using the MMA9550 Intelligent
Accelerometer to Detect the High
Point of a Vertical Trajectory
by Fengyi Li, Applications Engineer
1
Introduction
One of the classic applications of an accelerometer sensor is
the detection of the freefall condition when the sensor is no
longer supported and is following a ballistic trajectory under
gravity. If a notebook PC with an integrated accelerometer
sensor is held and then dropped without spinning, the
accelerometer reading falls from 1g to 0g at the moment the
notebook leaves the owner's hands and remains zero until
impact with the ground. Similarly, if the notebook is thrown
into the air without spinning, the accelerometer reading
again drops to zero once the notebook leaves the owner's
hands and remains zero until impact. In the presence of
spinning, a centripetal acceleration will be present which
complicates matters and prevents the accelerometer reading
reaching zero.
This application note addresses the related problem of how
to detect the high point (or apogee) of a vertical trajectory in
the absence of spinning. The instantaneous accelerometer
reading cannot be used because the reading is zero from the
moment it leaves the thrower's hands until the point of
impact with the ground. A more advanced approach is
required which integrates the accelerometer reading to
predict the high point of the trajectory where the vertical
© 2012 Freescale Semiconductor, Inc. All rights reserved.
Contents
1
2
3
4
Introduction . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . .
Mathematical Model . . . . . . . . . . . . . .
MMA9550 Implementation . . . . . . . . .
1
2
2
5
velocity is, by definition, zero. The Freescale MMA9550 smart accelerometer sensor, which contains a
processor and memory, is ideally suited for problems such as this. Reference C code to implement the
algorithm on the MMA9550 is therefore included with this application note (See Example 1).
The following Freescale Application Notes describe related topics that may be useful to the reader:
• Tilt Sensing Using Linear Accelerometers (document number AN3461), describes the use of
accelerometers for tilt estimation and portrait or landscape selection.
• High Precision Calibration of a Three Axis Accelerometer (document number AN4399), describes
how a product containing a consumer grade accelerometer can be re-calibrated after manufacture
to achieve a high level of accuracy.
2
Summary
In the absence of spinning, an accelerometer sensor which is dropped or thrown will report near zero g
acceleration from the moment it is released from the thrower's hands to the point it impacts the ground.
The instantaneous accelerometer reading cannot therefore be used to determine the high point of the
trajectory.
Integration of the accelerometer reading minus 1g during a vertical throw provides an estimate of the
accelerometer vertical velocity at the point of release. The moment of release is easily determined by the
appearance of near zero accelerometer readings.
The high point of a vertical trajectory has zero vertical velocity. The known 1g deceleration under gravity
allows the calculation of the time interval from release to the high point of the trajectory from the ratio of
the release velocity to 1g.
The Freescale MMA9550 smart accelerometer sensor is an ideal single device solution to perform the
accelerometer integration. Example 1 includes reference C code for an MMA9550 implementation which
flashes an LED at the top of the vertical trajectory.
3
Mathematical Model
Application Note AN3461 derives the reading G p of an accelerometer sensor oriented at an arbitrary angle
in the earth's gravitational field g and undergoing linear acceleration a r ( t ) (as measured in the earth's
reference frame r) as:
⎛ G px ⎞
⎜
⎟
G p = ⎜ G py ⎟ = R ( t ) { g – a r ( t ) }
Eqn. 1
⎜
⎟
⎝ G pz ⎠
Where R ( t ) is the rotation matrix describing the instantaneous orientation of the sensor relative to the
earth's reference frame.
At the moment that the accelerometer is released from the hand, it follows a ballistic freefall trajectory
where a r ( t ) = g . Equation 1 then simplifies to:
G p = 0 in freefall
Eqn. 2
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
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Freescale Semiconductor, Inc.
The detection of a zero, or near zero, accelerometer reading is therefore indicative that the accelerometer
has left the user's hands.
For the particular case where the accelerometer sensor is held at an arbitrary, but fixed, orientation and
accelerated in the vertical axis z only, Equation 1 simplifies to:
⎛ G px ⎞
⎛
⎞
0
⎜
⎟
⎜
⎟
G p = ⎜ G py ⎟ = R ⎜
Eqn. 3
⎟
0
⎜
⎟
⎜
⎟
⎝ G ⎠
⎝ g – ar ( t ) ⎠
pz
Where g is the magnitude of the earth's gravitation field or approximately 9.81 ms-2. The term a r ( t ) is
now the acceleration in the vertical z axis with the sign convention that an acceleration downwards parallel
to gravity has a positive sign.
The orientation matrix R can be eliminated from Equation 3 by taking the modulus of both sides to give:
Gp =
G px2 + G py2 + G pz2 = g – a r ( t )
Eqn. 4
The application we are considering is the vertical throw of the accelerometer and the detection of the high
point of the trajectory. The sequence of events will be:
1. The accelerometer will first be held steady with a r ( t ) ≈ 0 .
The magnitude of the accelerometer reading G p will therefore be approximately 1g.
2. When the hand holding of the accelerometer is lowered prior to the upwards throw, the linear
acceleration a r ( t ) will spike in the downwards (positive) direction, reduce towards zero, and then
spike in the upwards (negative) direction when the hand stops in the lowered position.
The magnitude of the accelerometer reading G p in Equation 4 will therefore spike below 1g,
return to 1g and then spike above 1g.
3. The vertical throw will start with a strong acceleration a r ( t ) in the upwards (negative) direction.
The magnitude of the accelerometer reading G p in Equation 4 will therefore increase
significantly above 1g.
4. From the moment of release until impact with the ground, the accelerometer is in a freefall ballistic
trajectory and a r ( t ) = g .
The magnitude of the accelerometer reading G p in Equation 4 is then zero until impact.
Figure 1 illustrates this sequence of events.
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
Freescale Semiconductor, Inc.
3
Impact
| Gp |
Upwards throw
Upwards throw
Release t1
Hand lowered
Resting
Resting
1g
Apogee t 2
0g
t0
Δt
t1
Time
t2
Figure 1. Magnitude of accelerometer output during throw
It is now necessary to impose one more constraint on the motion of the accelerometer. Because the release
of the accelerometer from the hand will be detected by the zero g freefall condition, it is necessary to
require that the downwards linear acceleration of the hand prior to the throw is always less than 1g so that
the zero g freefall condition in Equation 4 is never reached accidentally prior to the throw. In practice this
is easy to achieve using a smooth lowering of the hand rather than a jerky motion.
With this assumption, the modulus sign in Equation 4 can be removed giving:
Gp =
G px2 + G py2 + G pz2 = g – a r ( t )
Eqn. 5
Equation 5 can now be integrated to give the vertical velocity v ( t 1 ) at the moment of t1 when the
accelerometer is released from the hand at the end of throw. Where v ( t 1 ) is positive in the downwards
direction because of the sign convention that the positive z axis is downwards.
v ( t1 ) =
∫t
t1
0
a r ( t ) dt + v ( t 0 ) =
∫t
t1
( g – G p ) dt + v ( t 0 ) =
0
∫t
t1
( g – G px2 + G py2 + G pz2 ) dt + v ( t 0 ) Eqn. 6
0
If we identify the time t 0 (the start of the velocity integration) as being any time when hand holding the
accelerometer is in the lowered position then, by definition, v ( t 0 ) = 0 . This vague definition of t 0 is
acceptable because the modulus of the accelerometer reading equals 1g throughout the period when the
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
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Freescale Semiconductor, Inc.
hand is lowered and the integrand is zero throughout this period. The integrand only becomes non-zero
when the hand starts to accelerate upwards at the start of the throw.
The integration interval t 1 – t 0 will be set to a fixed value Δ t which is long enough to capture the
acceleration of the throw however not so long that it exceeds the period when the hand is held steady prior
to the throw. A value of approximately 0.5s has been found suitable. Equation 6 can then be written as:
v ( t1 ) =
∫t
t1
1
– Δt
( g – G px2 + G py2 + G pz2 ) dt
Eqn. 7
The time t1 when the accelerometer is released from the hand is easily determined by the freefall condition
G p ≈ 0 . It is now simple to integrate from the release time t1 to the time t2 at the apogee or high point of
the trajectory when the vertical velocity is zero.
v ( t2 ) = 0 =
t2
∫t
1
–v ( t1 )
⇒ t 2 – t 1 = -------------- =
g
g dt + v ( t 1 ) = g ( t 2 – t 1 ) + v ( t 1 )
∫t
t1
1
⎛ 1---⎞ ( G 2 + G 2 + G 2 – g ) dt
px
py
pz
⎝ ⎠
– Δt g
Eqn. 8
Eqn. 9
Equation 9 is the physically intuitive result that the time interval t 2 – t 1 from the release of the
accelerometer from the hand to reaching the high point of the trajectory is simply the upwards vertical
velocity divided by the acceleration due to gravity.
4
MMA9550 Implementation
The continuous time integral in Equation 9 must be replaced by a summation for a discrete time software
implementation. If the accelerometer is sampled at frequency fs, Equation 9 can be re-written for sample
number n as:
M–1
N =
1
∑ ⎛⎝ --g-⎞⎠ (
2
2
2
G px [ n – i ] + G py [ n – i ] + G pz [ n – i ] – g )
Eqn. 10
i=0
Where M is now the number of sampling intervals from the detection of the low g condition at release to
the accelerometer reaching the top of the trajectory. The number of terms M in the summation should be
selected so that Mf s ≈ Δt .
The sampling frequency for the MMA9550 implementation was selected to be 122 Hz. The integration
interval Δt was selected to be 64/122s = 0.52459s requiring a delay line length of 64 elements to store the
components of the sum in Equation 10.
The key code kernel for a C code implementation on the MMA9550 is listed Example 1.
Example 1. Reference C code
// accelerometer scaling constants
#define ONE_G 4096
// 1g on MMA9550 equals 4096 bits
#define ZEROP3_G 1229
// 0.3g (low g threshold) equals 1229 bits
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
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5
// interval constants for 122Hz sampling rate
#define ZEROP5246_S 64
// 0.5246s integration interval
#define ZEROPTWO_S 24
// 0.2s interval for LED flash
// acceleration delay line size
#define DLSIZE ZEROP5246_S
// 0.5246s or 64 elements delay line size
// local variables
int i;
int apogee_counter;
int DL_counter;
int gx, gy, gz;
float accel_mag;
float DLSum;
float DL[DLSIZE];
//
//
//
//
//
//
//
loop counter
countdown counter for LED flash at apogee
delay line index
accelerometer channel data (bits)
accelerometer magnitude (bits)
sum of delay line entries
delay line of normalised acceleration magnitudes
// zero the delay line
for (i = 0; i < DLSIZE; i++)
DL[i] = 0.0F;
// set apogee countdown timer to invalid value -1
apogee_counter = -1;
// zero the circular delay line index
DL_counter = 0;
// loop forever
while (TRUE)
{
// idle till sampling interrupt
idle();
//
gx
gy
gz
read the accelerometer data scaled 1g=4096 bits
= get_accel(XAXIS);
= get_accel(YAXIS);
= get_accel(ZAXIS);
// calculate accelerometer magnitude in bits
accel_mag = (float) sqrt(gx * gx + gy * gy + gz * gz);
// store the normalised linear acceleration in the circular delay line
DL[DL_counter++] = (accel_mag - (float) ONE_G) / (float) ONE_G;
// wrap the delay line index back to zero if necessary
if (DL_counter == DLSIZE)
DL_counter = 0;
// test for new release from hand using 0.3g as criterion
if ((apogee_counter < 0) && (accel_mag < ZEROP3_G))
{
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
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Freescale Semiconductor, Inc.
// sum the delay line to get sampling interval count to top of trajectory
DLSum = 0.0F;
for (i = 0; i < DLSIZE; i++)
DLSum += DL[i];
apogee_counter = (int) fabs(DLSum);
}
// if the apogee counter has reached zero, flash the LED for 0.2S
if (apogee_counter == 0)
led_pulse(LED_WHITE, ZEROPTWO_S);
// decrement the apogee counter if non-negative
if (apogee_counter >= 0)
apogee_counter--;
}
Using MMA9550 to Detect High Point of a Vertical Trajectory Application Note, Rev. 0
Freescale Semiconductor, Inc.
7
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© 2012 Freescale Semiconductor, Inc. All rights reserved.
Document Number: AN4446
Rev. 0
08/2012