FIXED POINT VERSUS FLOATING POINT MATHEMATICS IN

EMBEDDED SYSTEM PROGRAMMING FOR FLUID POWER

MECHATRONIC COMPONENTS CONTROL: A REAL CASE STUDY

Massimiliano RUGGERI*, Matteo FRACASSI**, Massimo MARTELLI* and Massimo DIAN*

* IMAMOTER, Institute for Agricultural and Earthmoving Machines

National research Council of Italy

Via Canal Bianco 28, Ferrara, 44124 Italy

(E-mail: m.ruggeri@imamoter.cnr.it)

** Research and Development Dept., Tecnord S.r.l. Company

Via Malavolti 36, Modena, 41100 Italy

ABSTRACT

The increased systems complexity and performance request for electro-hydraulic applications, ask for more performing

electronic systems and control functions. The new powerful microcontrollers and efficient cross compilers, encourage

the floating point mathematics usage in the software control routines, useful to directly reuse the routines generated by

the simulation tools, despite the lack of control for precise resulting routine execution.

The paper describes a practical experience of system performance optimization on a microcontroller installed on

electro-hydraulic systems for mobile applications. A deeper analysis carried out on execution time occupied by floating

point mathematic operations, working on the software side of the mechatronic component, led to a considerably better

performance. Here it is demonstrated that, without lack of precision, fixed point mathematics are more performing, if

executed by modern microcontrollers, even if more instructions are executed by the software routines due to the

necessary rescaling of factors needed by the requested precision.

KEY WORDS

Floating Point, Fixed Point, Electronic Valve Control, Embedded Systems, Real Time

NOMENCLATURE

I : current (A)

ms : milliseconds

ns : nanoseconds

P : Pressure (bar)

µs : microseconds

Q : Flow rate (l/min)

SP : Spool Position (mm)

V : Voltage (V)

INTRODUCTION

In the last years the complexity of digital control of

electro-hydraulic components and systems increased, due

to the continuous request for better performance,

efficiency and safety. Moreover, due to the increased

quality and reliability of the simulation environment the

electronic control of Mechatronic systems is moving

away from maps and vectors towards a model based

approach, in order to model in a better way the

controlled actual systems and to reduce the dependency

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on control parameterization. In this scenario, useful tools

allow to generate in various automatic ways C/C++

language code or pseudo-code, that can be transferred

into embedded systems compiled binary files, or in

hardware-in-the-loop platforms (HWIL), in order to test,

on the physical bench or directly in the machine, the

designed control strategies. This productivity increment

brings with it a design habit: the floating point

mathematic usage. Floating point mathematic is a

powerful method to calculate and to perform the control

actions, once the system is simulated or treated by a

HWIL powerful platform, but very difficult to be

managed by an embedded microcontroller not featuring a

Floating Point Unit (FPU). Sometimes it can be though

that a limited usage of Floating Point mathematic in a

software program doesn’t affect its real-time; this paper

demonstrates that this is normally a risky approach, due

to the high CPU load resulting from floating point to

fixed point conversions and vice-versa and due to the

uncertainty related to time required by the floating point

/ fixed point data conversion. A real case study of a

product whose control was optimized, and its

performance increased, is presented, analyzing the

performance and performance comparison from the

floating point mathematic to fixed point mathematic used

in two versions of embedded software in the mechatronic

component electronic control unit.

THE SYSTEM

The paper describes a real case study of a Two Stage

Directional Valve Spool Position Control.

The directional valve is a CAN controlled remote of an

agricultural tractor and the spool position is controlled by

a couple of three-way flow regulation valves,

electronically and independently controlled on the basis

of the spool position feedback from a contactless

position sensor. As shown in Figure 1, the two

proportional flow regulation valves are controlled by two

electronically controlled PWM signals, regulated by a

Freescale HCS12 series microcontroller, a powerful

16-bit core microcontroller widely used in mobile and

automotive applications. The only feedback related to the

position of the two proportional flow regulation valves is

the current feedback acquired through the Analog to

Digital Converter (ADC) of the microcontroller, while

the current regulation is realized through an hardware

circuitry, directly controlled by the PWM signal, by the

specialized output of the microcontroller. The two

proportional flow regulation valves are totally

independent, and can be separately controlled, in order to

increase the spool dynamic control performance. This

first stage valve configuration gives the possibility to

reduce the spool speed during transients, opening the

opposite valve in respect to the desired direction of the

spool movement, in order to avoid spool position

overshoot, that is not desirable in a flow control of

actuators. In fact a second stage spool position overshoot

can result in a damage in some applications, due to a

possible wrong actuator final position caused by an

excess of oil flow in the final stage of actuators (e.g.

hydraulic cylinders). The control module that is studied

in the paper is composed by the two first stage valves,

the electronic control unit embedded in the valves

enclosure and the second stage valve spool position

sensor. The module, called Multidrom®, is produced by

Tecnord Company and is mounted in a wide number of

directional valves and widely used both in agricultural

machines and in earthmoving and construction machines,

all over the world. The Multidrom® module adaptability

is a challenging characteristic because of the control

stability and robustness for the entire range of

applications.

In fact the control is the same for all applications, and the

tuning is realized only through a different parameter set

for each different application and valve section.

Figure 1 The Multidrom System Structure with

electro-hydraulic valves and the flow regulation spool

Figure 2 The Multidrom Module coupled with a

Directional Valve

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THE STATE OF THE ART

The Multidrom module is produced in various versions

and it is provided with a complex control strategy that is

based on a feed forward function implementing the

model of the electro-hydraulic flow control valves and

the second stage flow regulation valve; moreover, in

order to compensate the model uncertainties and the

environment variables, a feedback function based on a

PID with variable gains and a differentiated anti-windup

strategy is implemented, coupled with the feed forward

function. The PID was optimized on a test bench and an

important dependency from oil temperature was

observed. Thus, in order to reach a good compromise

between control stability and performance various

parameter sets were defined, as a function of the second

stage configuration. From the software point of view, all

strategies, implemented in standard C language, were

written using floating point mathematics, allowed by the

Metrowerks CodeWarrior compiler and by the Freescale

microcontroller used for the application. That

mathematics setting is a modern choice, allowed by the

increased calculation power of the new microcontroller

products on the market, protects from some kind of data

type error, leaving the control for data dimension to the

microcontroller, while implies a considerably greater

time to execute the requested operations with the

Arithmetic Control Unit (ALU) of the Microcontroller.

The resulting maximum rate for the control function

execution and the necessary controls for diagnosis and

communication was 5 ms (cycle period). This 200 Hz

control frequency is apt to control a classic second stage

dynamic of the flow regulation valve coupled with

Multidrom modules, whose step response was measured

to be 110 ms from central neutral position to the

complete open position both in extend and retract

direction, as shown in Figure 3; but the control dynamic

can result unable to take advantage of the higher

dynamics of the electro-hydraulic controlled valves of

which is provided the Multidrom module. Tests

performed on a test bench demonstrated that, in some

high dynamic working condition, the control was unable

to avoid flow regulated valve spool position overflow

and that the steady state error was difficult to be

nullified; in fact the control reactivity, needed for

dynamic performance, created a bigger reaction on small

dynamics and near the steady state spool position (in

Dark-Green in Figure 3). This is the classic

demonstration of the limits of linear controls, like the

PID (Red, Light-Green and Blue lines respectively in

Figure 3) even if provided with variable gains and

anti-windup system, that is unable to be tuned for highly

non-linear systems. The main idea of this work is to

reduce the CPU control time in order to increase the

frequency of the control function execution, in order to

observe the dynamics of the controlled systems in a

shorter time period, where the model linearization is

allowed and then the linear controller is apt to regulate

the system behavior.

THE CONTROL IMPLEMENTATION

The control strategy was primarily designed using a

mathematical model of the system, and, in a second

phase, was translated in ANSI C language and

cross-compiled for the HCS12 Freescale microcontroller

family using the CodeWarrior C/C++ compiler.

Many calculations are executed using single precision

float data type, 32 bits, 8 for the exponent, 23 for the

significand and 1 for the data sign as stated in [1] and [2],

that are expressed in the form:

( ) ( )12723

1

2211 −

=

− ⋅

⋅+⋅−= ∑ e

i

iibvalue

sign

(1)

Where sign is the most significant bit, bi are the base

fractional binary values and e is the exponential binary

value. The microcontroller converts all data using Eq. (1)

for each control function run using shift, multiplier and

divider functions of the ALU (Arithmetic Logic Unit)

both for direct and for reverse conversions.

For each addition/subtraction the ALU converts the

floating point numbers in order to obtain the same

exponent for all operation’s elements and then execute

the addition or subtraction; after the operation another

shift can be executed in order to normalize the result in

case of overflow or underflow of the operation executed;

similarly for multiplication and division. To multiply

for example, the significands are multiplied while the

exponents are added, and the result is rounded and

normalized. Thus, the floating point mathematic need for

more instruction to be executed by the microcontroller

ALU, in order to normalize the numbers before and after

the operation over the mantissa in fixed point mode.

Figure 3 Step Response with Floating Point Software

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The resulting control function task execution time can

vary considerably from the best to the worst case from

550 µs to 820 µs, depending on the number of floating

point operations that need shift for normalizing data to

the standard format. Figure 4 shows the multiple

acquisition of the control function execution time,

obtained using an Oscilloscope with the “persistency”

function activated at 2 seconds of period. The signal

shown is the status of an hardware output pin of the

Freescale microcontroller activated in the first instruction

of the control function and deactivated in the last

instruction of the control function; the task execution

time is then affected by the introduction of two new

instruction for switching the hardware pin status: the

time for each switch is around 25 ns at 80 MHz clock.

The real need for floating point was evaluated as

suggested in [3], and it was found that, using integer

numbers with 32 bit precision in the intermediate

computations of the control function, the range of values

was sufficient to cover the entire range of intermediate

calculations of the actual control implementation (also

solved in [4]). In fact fixed point mathematics does not

reduce the precision with respect to single precision float

data, if the value of variables does not change for more

than 7 orders of magnitude if, when necessary, 32 bit

variables are used; nevertheless the microcontroller

executes faster the control strategy.

Moreover a fixed point algorithm requires the designer to

write much more controlled code, because of the need to

verify the absence of overflows in fixed point operations,

while in floating point mathematics the designer can lose

the feeling on the data dimension in the control function

implementation.

Last but not least, considering the embedded systems

structure and method to design, tune and test digital

control systems, it is clear that a large number of floating

to fixed point and vice-versa are necessary, in order to

perform the complete software structure. Data to be

transmitted to a data logger over CAN network, for

example, are converted from floating point to fixed point,

in order to be sent in a readable format to the logger

through CAN messages; in the same way, control

parameters are sent and stored in memory in fixed point

format, as usual for human thinking. Moreover looking

at the sensor inputs and at the output of the control

system and analyzing the microcontroller peripheral

structure, the spool position is acquired by a 10 bit ADC

channel and the control action is actuated through a

PWM signal by a peripheral where a control register

must be filled with a 10 bit duty cycle value in real time.

All these values are in fixed point format and a

conversion is necessary to pass from/to floating point

format in control function.

Figure 5 Task Control Duration with Floating and Fixed

Point Mathematics

All these conversions and the strategies required by the

application limits the repetition time for the control task

at 5 ms at least. The mean value of the pure control

function time occupation with floating point mathematics

is shown in the Figure 5 above and is around 550 µs.

The new control was then implemented, modifying the

software control code by replacing all floating point

computation and variables with equivalent fixed point,

thus reducing the data dimension to 16 bit integer, when

possible, and using 32 bit variables where needed for

computation requirements.

The result for the mean value of the control function time

occupation with fixed point mathematics is shown in the

Figure 4 Uncertainty in Floating Point Control Task

Duration

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Figure 5 below and is around 180 µs.

This performance increase was similarly obtained in the

other functions of the application, where floating/fixed

point conversions were operated.

The result in terms of occupation time by the main

control function is shown in Figure 6, where in the above

image is presented the result for control action time

distribution at 5 ms, while in the below image the control

function time distribution at 1 ms is shown. The figures

demonstrate that no delay due to other tasks is present in

the control function time distribution and then that the

control is executed at deterministic time intervals. In

order to ensure the equivalence of the new software a test

version was created where, for each computation

modified both versions were implemented and a control

was added in order to evaluate both the consistency of

the new software version and the precision.

ResultFloatX = float_functionX();

ResultFixedX = fixed_functionX();

if ( ResultFloatX != (float) ResultFixedX )

{

error_management_function();

}

After the new and old versions comparison, the final

software version was settled at 500 µs of cycle time for

the main control function, resulting in a 10 factor for

control dynamic in respect to the original control.

This software configuration increased considerably the

dynamic and steady state system performance.

COMAPRATIVE TESTS AT TEST BENCH

A Multidrom equipped flow regulation valve was

installed at test bench and tested in various working

conditions, using both software versions reloaded on the

same unit for each test. The monitoring of control task

execution time and cycle time was maintained using a

probe directly connected on a software controlled pin of

the microcontroller (the red cable in the Figure 7), while

all other data, both from sensors and from control

corrections, were acquired through the Multidrom CAN

network with Vector Canalyzer at 500 µs frequency.

The same CAN network analyzer was used to simulate

CAN commands to the module through a dynamic script

that simulated time histories of set point for flow

regulation commands automatically sent by the same tool.

Figure 6 Task Control Period and precise repeatability

at 5 ms and 1 ms timing (new software)

Figure 7 The System under test at the test bench

Figure 8 Step full displacement response with control

contribution and Peak strategy: Integral term Blue line,

D term Green line, P term light blue line.

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Figure 8 shows the main control strategy behavior, that

applies a fully powered command for big errors in set

point spool position, while starting to use the variable

gains PID for small errors [5].

In the figure a full extend to retract transient is presented,

in order to show the second stage spool dynamics.

A very small spool position overflow is observed and it

can be noted that it does not result in a higher flow

because it is in the fully opened spool displacement

region.

Figure 9 Small step response

The green line is the derivative correction, the light blue

is the proportional correction (saturated) and the blue is

the integral term correction. The values in the vertical axis

are the ADC acquisition values in 10 bit resolution. The

total spool displacement is 16 mm (at 0,02 mm/bit).The

best results are in the small set-point variations and in

steady state error, that results stable with the new control.

As show in the comparison in the two graphics in Figure

9 for small step response, here 0,35 mm, the faster control

action results in a 10 times reduced time for reaching the

90 % of the difference and absence of overshoot (new

software with fixed point mathematics: below graphic;

old software with floating point mathematics: above

graphic). In both graphics the green line is the set-point

for spool position and the blue line is the spool position

direct acquisition through the Hall effect sensor in the

Multidrom module.

Figure 10 Sinusoidal Set-point with Step dynamic

As presented in Figure 10 a variable set-point with steps

response and a sinusoidal waveform results in a better

dynamic response in the new software (below graphic in

the Figure) and in a uncorrected spool position error in the

old software version (above graphic in the Figure), while

the dynamic response at big steps are comparable for the

two software versions.

In both graphics the green line is the set point command

and the blue line is the spool position sensor signal.

The same differences can be found in Figure 11, where

step response coupled with a saw-tooth set-point time

history was generated (new software in the below

graphic in the Figure). The most remarkable result is the

control robustness in function of the control pressure that

should be optimal at 20 bar, but that can be varied in a

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range of values without lack of precision in flow

regulation, except for the dynamic performance.

With the new software version the Multidrom module

allows the control pressure reduction to 8 bar, while the

original software is unable to reach the set-point value

and the diagnostic strategies lead it to protection condition

in safe state (neutral spool position).

Figure 11 Saw-Tooth Set-point with Step dynamic

As shown in Figure 12 a complete retract to extend

transient was generated with low dynamics in 30 s time,

in order to characterize the steady state error with 8 bar of

control pressure in the Multidrom flow control valves.

In the above graphic in the Figure, it can be noted that the

old software strategies, based on Floating point

mathematics, are unable to reach the set-point and, after a

diagnosis time the control valves are switched off by the

module, in order to lead the spool of the controlled vale in

the neutral position (safe state).

Conversely in the below graphic in the Figure, it can be

observed the entire stationary characteristic of the spool

position control response, that is lead by the new software

based on fixed point mathematics, through the neutral

position at 400 ADC value, that corresponds to 8 mm

(0,02 mm/bit for the spool position acquisition), with no

error in steady state control even if in low (degraded)

pressure condition. In the graphic are also in evidence the

two dead-band and the two polarization currents near by

the valve center rest position.

A higher dynamic without spool position overflow, the

absence of steady state error, the complete spool position

hysteresis and an increased robustness against the control

pressure variation are the most remarkable results; all

these performance increase are due to the PID feedback

correction, that results easier for smaller errors that are

observed in shorter time between two control actions.

CONCLUSIONS

The optimization of the control function, obtained with

the sole mathematics optimization performed on a real

product case, represents an important case study for all

research groups that need for more performance for an

embedded control system.

The limited computing power available in the embedded

systems and microcontrollers, even if in continuous

Figure 12 Low pressure quasi-static characteristic

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growth, is not sufficient to implement floating point

mathematics based control functions, obtained for

example as output products of widely used simulation

tool-suites.

Conversely fixed point mathematics, the standard

embedded mathematical library in the automotive sector

today, is suitable for optimal control and fits the data

field for the most part of applications without lack of

precision in respect to floating point mathematics and

represents a good compromise for real time control of

mechatronic components in electro-hydraulic

applications.

The next step for fixed point mathematics test and

validation, in our study for electro hydraulic controls

design, will be the direct synthesis of fixed point control

blocks in simulation environment and their test on

models and in control function translation for embedded

systems programming [6]. This method should allow a

direct comparison on control performance compared

with floating point mathematics control blocks and

functions directly on simulation environment.

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1. IEEE 754, ed. 2008: IEEE Standard for Binary

Floating-Point Arithmetic, IEEE

2. IEC 60559 ed. 2.0: 1989, Binary floating-point

arithmetic for microprocessor systems, IEC, Swiss.

3. James W. Gray, PID Routines for MC68HC11K4 and

MC68HC11N4 Microcontrollers, AN1215/D

Freescale Semiconductor Inc., 2004.

4 . Elmenreich W., Rosenblatt M., Wolf A., Fixed Point

Library Based on ISO/IEC Standard DTR 18037 for

Atmel AVR Microcontrollers, 5th Workshop on

Intelligent Solutions in Embedded Systems, June

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IEEE Transactions on Circuits and Systems II, Vol.

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