In the device where the compliance is planted inherently, the encoder installed on the motor cannot give the exact position of the jaw on the occasion of activating compliance; while the actual position of the jaw indicates the actual position that is essentially put into the position feedback loop and the higher control loop as an key parameter to tune the system to the desired position. The position of the jaw can be obtained by either directly a sensor installed on the jaw or indirectly calculation from the monitored crank angle. When the device starts from the initial position, with the crank rotating, the mouth increasingly opens, while the angle between the initially horizontal tangent at the bottom point of the chin and the ground is varying monotonously till the opening limit. Hence, the chin angle denoted as ߛ is seen to be an optimal parameter revealing the position of the jaw with respect to the crank angle in this application. The chin angle is actually the varying angle of the coupler in a rigid-body guidance based mechanism. By referring the same denotation, the angular displacement of the coupler ߠଷ is expressed as the function of the crank ߠଶ.
The angular displacement on the chin is implemented by a tilt measuring unit, which comprises of a dual-axis accelerometer ADXL203 and two cascaded amplifiers; 9ADXL203 can measure both static and dynamic acceleration of two axes that are normal to the direction of the gravity upon the high performance of sensitivity accuracy. General technical characteristics are listed in Table 4-3, where non-relevant specifications in this application are
9 More information can be referred to the product manual, please visit: http://www.analog.com/static/imported- files/data_sheets/ADXL103_203.pdf.
not given. The tilt measurement utilizes the vector of the gravity as an input acceleration, which projects on the sensing axis produces output voltage component in relation with the angle between the sensing axis and the horizon (parallel to the earth). The dual-axis accelerometer is laid to place two sensing axis in the plane where the tilt occurs as shown in Figure 4-13, so both axes are involved in the measurement of the same tilt angle, i.e. the coupler angular displacement. The angle between one axis and the horizon increases while the other decrease. The sensitivity to angular variance stays highest at the orientation of 0g reading on axis parallel to the ground and shrinks when the angle is increasing. The sensitivity of either axis can be mutually compensating.
The output acceleration of both axes referring to X-axis and Z-axis can be calculated in Eq.(4-1).
൜ܣܣ ൌ ͳ݃ ൈ ߠ
ൌ ͳ݃ ൈ ߠ (4-1)
Table 4-3 Specification of the ADXL203
Sensor Properties
Size 5 mm × 2 mm × 2 mm
Package LCC
Power supply 5 V (operating range 3V-6V)
Shock survival 3500 g Output 1.5 V-3.5 V TypicalPerformance Full-scale range ± 1.7 g Sensitivity (maximum) 1000 mV/g 0g Voltage 2.5 V
Initial 0g output deviation ± 25 mg
Resolution 1 mg at 60 Hz (approximately 0.06° of inclination)
Axis alignment ≤ 0.1°
Figure 4-13 Two axes application (adapted from product manual)
The output of the accelerometer is analog voltage, theoretically ranging from 1.5 V to 3.5 V. A built-in ADC in MCU is converting analog information to the digital data readable with input range of 0 to 3.3 V. According to the manual the product has a bias of 100 mV shift of zero-g
and 40 mV/g shift of sensitivity; the worst-case output voltage is ranged within 1.36 V to3.64V, which is shifted to fit the input ADC range via a two-stage amplifier circuit. The first stage amplifies the signal 1.2 times based on common-mode voltage of 2V, while the second one has a gain of 1.1 based on the common-mode voltage 1.7 V. The amplifier carried out by a quad AD8608 has a total gain of 1.32 and the output range is in well accordance to the input range. The amplified voltage corresponding to the output of accelerometer is expressed in Eq. (4-2).
ܸௗ ൌ ൫ܸെ ʹǤͷ൯ ൈ ͳǤʹ ൈ ͳǤͳ ͳǤ (4-2)
The maximum of amplified voltage and the minimum is calculated as 3.2 V and 0.2 V respectively so the output range of tilt system is 3 V. The resolution of the voltage per least significant bit (LSB) from a 10-bit ADC is expressed:
ܮܵܤ ൌ͵Ǥ͵ܸ
ʹଵ ൌ ͲǤͲͲʹͻܸ ൎ ͵ܸ݉ (4-3)
As specified, for the single-axis inclination measurement, the desired inclination resolution of the system derived from the controller precision is 0.4° for a range of 0° to 55°. The output acceleration is trigonometric function of the tilt angle, so the resolution of acceleration is variable, and it can be expressed in terms of the incremental sensitivity of a certain step size in Eq. (4-4). ȟ݃ ൌ ͳ݃ ൈ ሺ ߮ െ ሺ߮ െ ο߮ሻሻ ൌ ߮ െ ሺ߮ െ ͲǤͶιሻ ൌ ʹ ൬ʹ߮ െ ͲǤͶ ʹ ൰ ͲǤͶ ʹ (4-4) Where: ߮ is the current angle, ȟ߮ is the step size.
Therefore, the resolution of the acceleration is monotonously dropping while the angle is growing up. The largest resolution locates at ߮ ൌ Ͳι, ȟ݃ ൌ ݉݃ while the least point stays at ߮ ൌ ͷͷι, ȟ݃ ൌ Ͷ݉݃. The minimum resolution is chosen as the constant acceleration per LSB that can be achieved by single-axis measurement. The resolution of the output voltage per constant acceleration is calculated:
οܸ ൌ ͳͲͲͲܸ݉Ȁ݃ ൈ ȟ݃ ൌ Ͷܸ݉ (4-5)
As the output voltage of constant acceleration οܸ is much larger than the resolution ܮܵܤ that can be distinguished by ADC, the tilt unit is able to meet the requirement from controller. Since two axes that are orientated to in the plane where the inclination occurs are able to yield the acceleration, the effective incremental sensitivity stays roughly constant within the
range of all inclination angles. The resolution of the acceleration can be represented in Eq. (4-7) with respect to an inclination change δ°, i.e.
ȟܣ௨௧ ؆ ͳ݃ ൈ ߜ (4-6)
ߜ ൌ ିଵሺȟܣ௨௧ሻ ൌ ିଵ൬
ܮܵܤ
ͳͲͲͲܸ݉Ȁ݃൰ (4-7)
The allowed minimum acceleration resolution that can be differentiated by the ADC is calculated as 0.17°, far less than the resolution of the changing angle required by the system accuracy. Then the output voltage denoted as ܸሺǡሻ representing the measured acceleration along each axis in Eq. (4-8) on the identical axis is converted to the angle of inclination, which can be calculated in Eq. (4-9). Since the calculation basically presents the same value, it can also be expressed in Eq. (4-11) by easier approach of using the ratio of two values of Eq. (4-10).
ܸሺǡሻൌ ൫ͳͲͲͲܸ݉Ȁ݃ ൈ ܣሺǡሻ൯ ൈ ͳǤ͵ʹ ͳǤ (4-8) ൞ߠ ൌ ିଵ൬ܸെ ͳǤ ͳǤ͵ʹ ൰ ߠൌ ିଵ൬ܸെ ͳǤ ͳǤ͵ʹ ൰ (4-9) ߠ ൌܣ ܣ (4-10) ߠ ൌ ିଵ൬ܸെ ͳǤ ܸെ ͳǤ൰ (4-11)
Note: a low-pass filter of 50 Hz bandwidth specified is accessorily created along the output by attaching capacitors to the internal 32 KΩ resistors, which is supportive of ADC sampling rate of 200 Hz, and the capacitor of a -3 dB bandwidth is pointed from:
݂௨௧ൌ ͳȀ൫ʹߨሺ͵ʹ݇ȳሻ ൈ ܥሺǡሻ൯ (4-12)
Where:
ܥሺǡሻൌ ͲǤͳߤܨȁ݂௨௧
The tilt measuring unit shown in Figure 4-14 is designed to be detachable with interfaces to power supply and output. The circuit board is prototyped as similarly shaped to fit the installation area on the side of the device, given in Appendix Figure 8-1. From of whole assembled device, the circuit board is finely placed on the coupler hand to align one sensing axis at the initial position to the direction of the gravity as much as possible, and the PCB installation can be found in Figure 8-2.
Figure 4-14 Tilt measurement unit circuit