Thursday, March 08, 2001

 

Calculating Forces and Moments from Biomechanics Platform, and Transducer Outputs

 

                The Calibration Matrix provides the means to calculate the force and moment inputs to the platform (Fj) based on the platform outputs voltages (Vi).  The following document describes how to perform these calculations.

 

                In the Biomechanics Force Platform Calibration document delivered with each platform a Calibration Matrix may be found on the bottom of page 3 (in SI units) and on the bottom of page 4 (in USC units).  The matrix is titled Calibration Matrix C(i,j).

 

                In this document the Calibration Matrix will be referred to as C (bold C), elements of the matrix will be referred to as Cij where i is the row index and j is the column index.

 

                The Calibration Matrix is presented in the above mentioned document in the following format:

                                                               

 

 

 

Fx’

Fy’

Fz’

Mx’

My’

Mz’

 

 

 

 

 

 

 

 

 

 

 

Fx

C11

C12

C13

C14

C15

C16

 

 

Fy

C21

C22

C23

C24

C25

C26

 

 

Fz

C31

C32

C33

C34

C35

C36

 

 

Mx

C41

C42

C43

C44

C45

C46

 

 

My

C51

C52

C53

C54

C55

C56

 

 

Mz

C61

C62

C63

C64

C65

C66

 

 

 

 

 

 

 

 

 

                The labels across the top, Fx’ through  Mz’, indicate that the columns refer to the platform output channels.  The labels down the left hand side, Fx through Mz , indicate that the rows correspond to the input loads acting on the platform. 

 

                The calibration matrix is used in conjunction with the voltage outputs from the six channels of the platform.  These voltages can be represented by a one by six matrix Vi :

 

                                                V1

                                                                        V2

                                                Vi       =          V3

                                                                        V4

                                                                        V5

                                                                        V6

 

where V1 through V6 represent the output voltages from the six channels,  Fx, Fy, Fz, Mx, My, and Mz of the transducer respectively.

 

                The force and moment inputs to the transducer are calculated from the platform voltage outputs and the C matrix.  In matrix short hand this calculation is simplified to the following:

 

1)                             F = C · V

 

 

 

 

 

 

                Equation 1 above, when expanded, yields the following:

                                               

 

 

F1

C11 C12 C13 C14 C15 C16

   V1

 

 

F2

C21 C22 C23 C24 C25 C26

   V2

 

 

F3        =

C31 C32 C33 C34 C35 C36      ·

   V3

 

 

F4

C41 C42 C43 C44 C45 C46

   V4

 

 

F5

C51 C52 C53 C54 C55 C56

   V5

 

 

F6

C61 C62 C63 C64 C65 C66

   V6

 

 

 

 

 

 

                Again, the terms V1 through V6 represent the transducer voltage outputs corresponding to the Fx, Fy, Fz, Mx, My, and Mz  channels.

 

                Similarly, the terms F1 through F6 correspond to the force and moment loads, Fx, Fy, Fz, Mx, My, or Mz , acting on the platform.

 

                The ith force or moment input is calculated as the sum of the products of the six voltage outputs times the six elements of the ith row of the calibration matrix as shown below.

                                                                           6

2)                             Fi =          å  Vi · Cij

                                                                        j=1

                As an example, F1 which corresponds to Fx , would be calculated as follows:

 

3)                             F1 =         V1 ·C11 + V2 ·C12 + V3 ·C13 + V4 ·C14 + V5 ·C15 + V6 ·C16

 

                It is important to understand that the Vi outputs are the actual outputs of the platform (not the measured voltages from the amplifier) and that the elements Cij of the C matrix act on those actual platform outputs.  The C matrix is expressed in this manner to provide flexibility in accommodating various programming situations which will be discussed in the sections below but first a general discussion on the elements of the signal conditioning and data acquisition system is necessary.  We will first define the following terms:

 

                Excitation Voltage                Vexi                              (the excitation voltage)

                Gain                                        Gi                                           (the amplifier gain)

                Amplifier Output                  Vai                           (the voltage output of the amplifier)

 

                A strain gage bridge amplifier must be used in conjunction with AMTI’s force platform to provide excitation voltage for the sensor elements and to amplify the platforms output signals .  AMTI provides several models of amplifiers for this purpose.  Each of AMTI’s amplifiers has some method of setting the gain and excitation voltage although the methods differ depending on the amplifier model.  The gains and excitation voltages achieved by making the hardware selections on the amplifiers are essentially nominal values.  The actual calibrated gains and excitation voltages are specified for each channel and for each selectable level and are provided on a calibration sheet delivered with each amplifier.  Some of AMTI’s amplifiers allow the gains and excitation voltages to be set independently for each channel and other models limit excitation voltage choices to apply to all channels. 

 

                In the above definitions each of the variables correspond to a single column, six row matrix with elements that correspond to each of the amplifiers channels.  The amplifier channels are normally hard wired to correspond to the force and moment inputs, Fx, Fy, Fz, Mx, My, or Mz , acting on the platform.

 

Calculation when using voltage

                Some higher level programming environments designed for data acquisition (Lab View, MathCad and others) will make the output of the A/D card available in Volts.  If that is the case then the forces and moments may be calculated directly from the voltage data as follows.

                At a constant load, the output voltage of any channel of the platform will be directly proportional to the excitation voltage and gain.  To accommodate this the terms of the C matrix, Cij , have been normalized to an excitation voltage of 1 Volt and a gain of 1.  Thus the output voltage of the amplifier can be expressed as the following product of the excitation voltage, gain and platform output voltage.

 

4)                             Vai = Vexi  ·  Gi · Vi  · (1 · 10-6)

 

                Note that the 1 · 10-6 factor in the above arises because the elements of the C matrix are expressed in Volts-Engineering Units / micro-Volt and this term is applied to convert the units back into Volts.

 

                Reiterating, the C matrix acts on the output voltages of the platform (Vi).  Given the output voltages of the amplifier it is possible to determine the output voltage of the platform from the above equation as follows. 

 

5)                             Vi = Vai  / ( Vexi  ·  Gi · (1 · 10-6))

 

                The terms in the denominator of equation 5 above are all constants (but they may be different for each channel) and may be condensed into a single “gain factor” term which will be called GFi. 

 

6)                             GFi = 1 / ( Vexi  · Gi · (1 · 10-6) )

 

                Using the gain factor term equation 5 becomes:

 

7)                             Vi = GFi ·Vai

 

                Now equation may 7 be substituted into equation 2 above giving:

 

                                                                           6

8)                             Fi =          å  GFi ·Vai · Cij

                                                                        j=1

 

                Repeating the example in 3 above, based on the amplifier output voltages we have:

 

9)                             F1 =         GF1 · Va1 · C11 + GF2 · Va2 · C12 + GF3 · Va3 · C13

                                                 + GF4 · Va4 · C14 + GF5 · Va5 · C15 + GF6 · Va6 · C16 =                       Fx

                                                                                ·

                                                                                ·

                                                                                ·

                                F6 =         GF1 · Va1 · C61 + GF2 · Va2 · C62 + GF3 · Va3 · C63

                                                 + GF4 · Va4 · C64 + GF5 · Va5 · C65 + GF6 · Va6 · C66 =                       Mz

 

Calculation from  bits or count

                Lower level languages and programming environments have no built in “knowledge” of the ADC (analog to digital conversion) card and information describing this process must also be built into your program and calculations.  There are three things that you must know about the ADC card and it’s setup in your system.  These are:

 

                                · digital representation

                                · bit resolution

                                · maximum and minimum reference voltages

 

                The ADC card samples the analog voltages presented on each of it’s inputs and converts those voltages into some digital value.  These digital values must in turn be properly scaled and otherwise converted so that they numerically represent the voltages presented at the cards input (i.e. the amplifiers output voltages).

 

                The most commonly found digital representation used by ADC card manufacturers are the “offset binary” and the “two’s complement” representations.  You must check with your card manufacturer to determine their format and the required conversion process.

 

                The DT3002 card (the card that AMTI provides) uses the “offset binary” digital representation.  The offset binary representation represents data as unsigned values that range from zero to the positive maximum bit count of the card.

 

                The most commonly found bit resolutions for ADC cards are 12 and 16 bits.  A 12 bit card provides a maximum count of 2^12 or 4096, a 16 bit card provides a maximum count of 2^16 or  65536.

 

                The DT3002 card has a 12 bit resolution providing a maximum count of 4096.  Since the digital representation is in offset binary negative values will be represented by values from 0 to 2048 and positive values will be represented by values from 2048 to 4096.  These values are converted to a more convenient representation (two’s complement) by subtracting 2048 from each value.  The maximum and minimum bit counts will be called ADCmax and  ADCmin respectively.

 

                The ADC card converts the voltages presented on it’s inputs by comparing the voltage to two reference voltages.  Many cards can be set up for different reference voltages (this is often set by setting the full scale range of the card).  The DT3002 card may be set to have a full scale range of +- 10 Volts or +- 5 Volts.  We usually chose the +-10 Volt range.  The maximum and minimum voltage range will be called VRefmax and VRefmin respectively.

 

                Let the output values from the card be called Di where i represents the card channel.   If the ADC cards digital representation, bit range and voltage references are known the input voltage to the card may be determined.  Please note that the input voltage to the card is the output voltage from the amplifier (Vai) which we used in equations 7 and 8 above.  Vai is calculated as follows for a 12 bit card using offset binary representation:

 

8)                             Vaj  = (Dj – 2048) · (VRefmax- VRefmin) / ( ADCmax - ADCmin )

 

                Once Vai is known then equations 8 and 9 above can be used to determine the forces and moments acting on the platform.

 

Summary

                The calculations necessary to determine the forces and moments applied to the platform can be summarized into three steps.  If your software environment is “data acquisition card aware” and provides an output already converted to Volts then step 1 may be skipped and you may proceed with the step two calculations.

 

Step 1

                Calculate the voltage output of the amplifier for each channel as follows:

 

                                Vai  = (Dj – 2048) · (VRefmax- VRefmin) / ( ADCmax - ADCmin )

 

Step 2

                Calculate the “gain factor” for each channel:

 

                                GFi = 1 / ( Vexi  · Gi · (1 · 10-6) )

 

Step 3

                Determine the force and moment inputs as follows:

                                                                           6

                                Fi =          å  GFi · Vai · Cij

                                                                        j=1

 

Greatest Accuracy

                For the greatest accuracy the full calibration matrix should be used as described above.  This helps eliminate small errors due to transducer channel to channel cross talk.  Additionally the gains, Gi , and the excitation voltages, Vexi , for each channel should be obtained from the amplifier’s calibration sheet

 

Simplifications

                The above calculations can be simplified by using only the main diagonal terms of the C matrix as illustrated below.  The main diagonal terms in the matrix are highlighted in bold italic. 

 

                                                C11                C12                C13                C14                C15                C16

                                                                        C21                C22                C23                C24                C25                C26

                                                                        C31                C32                C33                C34                C35                C36

                                                                        C41                C42                C43                C44                C45                C46

                                                                        C51                C52                C53                C54                C55                C56

                                                                        C61                C62                C63                C64                C65                C66

 

                The off diagonal terms (which are not highlighted above) represent the cross talk terms of the transducer.  If all of the cross talk terms are assumed to be 0 then the matrix simplifies to:

 

                                                C11                0                     0                     0                     0                     0

                                                                        0                     C22                0                     0                     0                     0

                                                                        0                     0                     C33                0                     0                     0

                                                                        0                     0                     0                     C44                0                     0

                                                                        0                     0                     0                     0                     C55                0

                                                                        0                     0                     0                     0                     0                     C66

 

                The equation for calculating each force and moment channel becomes:

                                  

                                Fi =          GFi · Vai · Cjj

                       

and as an example the F1 force (Fx) would be calculated:

 

                                F1 =         GF1 · Va1 · C11

 

                It still remains necessary to calculate Vaj as described in equation 8 above.

 

                This simplification results in some loss of accuracy but with AMTI’s biomechanics platforms the cross talk terms are very small and if computational speed is of the greatest importance the simplification is justified.