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 2 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 4 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 Freescale Semiconductor, Inc. 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 6 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 How to Reach Us: Information in this document is provided solely to enable system and software Home Page: freescale.com implementers to use Freescale products. There are no express or implied copyright Web Support: freescale.com/support information in this document. licenses granted hereunder to design or fabricate any integrated circuits based on the Freescale reserves the right to make changes without further notice to any products herein. Freescale makes no warranty, representation, or guarantee regarding the suitability of its products for any particular purpose, nor does Freescale assume any liability arising out of the application or use of any product or circuit, and specifically disclaims any and all liability, including without limitation consequential or incidental damages. “Typical” parameters that may be provided in Freescale data sheets and/or specifications can and do vary in different applications, and actual performance may vary over time. All operating parameters, including “typicals,” must be validated for each customer application by customer’s technical experts. Freescale does not convey any license under its patent rights nor the rights of others. Freescale sells products pursuant to standard terms and conditions of sale, which can be found at the following address: freescale.com/salestermsandconditions. Freescale, the Freescale logo, AltiVec, C-5, CodeTest, CodeWarrior, ColdFire, C-Ware, Energy Efficient Solutions logo, Kinetis, mobileGT, PowerQUICC, Processor Expert, QorIQ, Qorivva, StarCore, Symphony, and VortiQa are trademarks of Freescale Semiconductor, Inc., Reg. U.S. Pat. & Tm. Off. Airfast, BeeKit, BeeStack, ColdFire+, CoreNet, Flexis, MagniV, MXC, Platform in a Package, QorIQ Qonverge, QUICC Engine, Ready Play, SafeAssure, SMARTMOS, TurboLink, Vybrid, and Xtrinsic are trademarks of Freescale Semiconductor, Inc. All other product or service names are the property of their respective owners. © 2012 Freescale Semiconductor, Inc. All rights reserved. Document Number: AN4446 Rev. 0 08/2012

- Similar pages
- ETC AN2848
- TI SLUA159
- Measuring Tilt with Low-g Accelerometers
- MOTOROLA MMA6261QR2
- Tilt Sensing Using Linear Accelerometers
- DEMOMMA8491 NXP Accelerometer Eval Kit
- HOLTEK HT1137A
- FREESCALE AFT21S230S_232S
- FREESCALE MPXV6115V_12
- FREESCALE MMA6255AEGR2
- VTI CMA3000-D01-30
- FREESCALE MRF8S7235N
- FREESCALE MRF8S18210WGHSR3
- NELLSEMI 50PT12SM