Getting my feet wet with Arduino Uno

July 18, 2013

Arduino Uno

I have tried to stay away from Arduino mainly because I didn’t want to learn a whole new culture. However, my AlgalitaROV project uses the OPENROV electronic architecture which employs a Cape that uses the Arduino Uno processor which in turn attaches to a BeagleBone Black(rats…another processor culture to learn).
Downloaded and installed the Arduino SDK 1.0.5 from and read through the getting started and reviewed the types of commands it uses. I bought a Arduino Uno starter kit from amazon ($32) which has a breadboard and some jumper wires. plugged in the Arduino Uno and it installed automatically without updating the drivers as suggested in the instructions. First observations:

1) the general purpose I/O (GPIO) are 5 volt at the breakouts. This is great since it is compatible with Vex parts. Current is o limited to 40ma per pin.

2) it can be powered from the USB cable or an external supply that must range from 6v to 20v with a recommendation of 7v to 12v.

3) Software wise in the Arduino you write a void setup() function and a void loop() function. These functions are called automatically by the operating system. setup() runs once and the loop() runs continuously…meaning the hidden main() function contains a while(true){loop()} structure that runs after the setup() runs. By contrast the Vex RobotC template calls a function pre_autonomous() once and executes separate tasks autonomous() and usercontrol(). usercontrol contains an explicit while(true){} which the programmer can change. Not sure that Arduino gives us that capability.

3)There are plenty of software examples and the language is close to C. The only function that seemed misnamed was analogWrite(pin,duty). This doesn’t write an analog output to a d/a hardware as one might assume but puts a 5 volt PWM signal output on a PWM capable GPIO pins. On the Uno 6 of the 14 GPIO pins are PWM capable (3,5,6,9,10,11). Not sure why they didn’t just call it PWMWrite().

4)Servo library is available for use but be careful with servo.write(int val).

Here is the function code from the servo library.  Note that if val is >544 then the command is assumed to be in microseconds rather than servo degrees.  Also note if it is less than 544 that the degree values are truncated to valid servo range of [0,180].    So a val = 255 will produce a 180 degree command.

void Servo::write(int value)
  if(value < MIN_PULSE_WIDTH)
  {  // treat values less than 544 as angles in degrees (valid values in microseconds are handled as microseconds)
    if(value < 0) value = 0;
    if(value > 180) value = 180;
    value = map(value, 0, 180, SERVO_MIN(),  SERVO_MAX());

First Script.
We need an r/c brushless motor driver. The ESC needs a standard servo drive signal. So I hooked up a Vex pot to the board and read its value [0, 1023] and scaled it to [0,180] which is given to a canned servo program object which has a servo.write which generates the proper servo output to a specified pin.  A motor is a continuous rotation servo  so 0 value represents max reverse speed and 180 value gives max forward speed.   I also included a serial printout of the value to the laptop console.
Hardware Setup:
[click on picture to expand to full size]

Shows breadboard wiring , motor 7.2v battery and external board 5 volt power supply

Shows breadboard wiring , motor 7.2v battery and external board 5 volt power supply

It is important to note that the motor is powered by a separate battery since it draws more current than the board can supply.

Closeup of arduino wiring

Closeup of arduino wiring

Full test setup

Full test setup


// Controlling a servo/motor position using a potentiometer (variable resistor) 
// by Chris Siegert:
// adapted from  Michal Rinott servo code  
// connect the center pin of a potentiometer (pot) to the potpin.  Connect the other two pins to the arduino 5v and ground pins.
// connect the white wire of the motor pwm cable to the servopin.  
// Important!! Connect the red and black wires to an external motor power source. 
// The arduino is not designed to deliver enough current to drive a servo/motor.
#include <Servo.h> 

Servo myservo;  // create servo object to control a servo or motor. 
int servopin = 9; //connects servo to pin 9 which is a pwm pin.
int potpin = 0;  // analog pin A0 used to connect the potentiometer
int val;    // variable to read the value from the analog pin 
int cmd = 0; // cmd to motor [0 to 180]
int cnt = 0; // program cycle counter 
int cnt_max = 33;   // this is the number of delay cycles between serial printouts. (15ms per cnt) 33 counts gives about 2 per second)  
void setup() 
  Serial.begin(9600);   //start up the serial data port at 9600 baud 
  myservo.attach(servopin);  // attaches the servo on servopin number to the servo object void loop() 
  val = analogRead(potpin);            // reads the value of the potentiometer (value between 0 and 1023) 
  cmd = map(val, 0, 1023, 0, 180);     // scale it in degrees of servo rotation(value between 0 and 180) 

  myservo.write(cmd);                  // sets the servo position according to the scaled value in deg 
  if(cnt > cnt_max)                  // this sets time between serial print outs to cnt_max cycles  or about (15 ms*cnt_max)
    cnt = 0;      // debug value
  delay(15);                           // waits for the servo to get there 

Note: Using a Line Sensor to Update Gyro Heading

February 28, 2013

Using Line Sensor to Update Gyro Heading

The programming challenge for the 2013 Sack Attack game was attempted by team 1508 LancerBots fully autonomous robot. It uses gyro information to generate a navigation solution and then automatically sequences through several way points to solve the route sequences. Unfortunately as mentioned in earlier posts, the gyro drift causes heading corruption which then causes sacks to by slightly missed. One possible solution I am proposing is to use line sensors to update the gyro.
Figure 1 shows the geometry and formulas required to accomplish the update for a special case when the line is parallel to the truth x axis.    The coordinates used by 1508 were specifically chosen so all lines were parallel to either the x or y  axis.  This simplifies the update equations.

Simply put… when a line sensor event occurs, note the position error  with respect to the line (using the Nav coordinates) and divide it by the range from the origin.    This ratio is the gyro drift error in radians.

Error Contributions:  

Errors in the update equation can stem from errors in  the Nav initial condition (<.25 in, 1 deg heading) , encoder scaling (typically <1%) , encoder resolution(< .03 in) , line location measurements (<.25 in), line tape width (<.5 in) and Navigation solution numerical errors (sampling, round off).

The dominant error is the encoder scaling which can add up when the encoder has moved several hundred inches.   Lines can also be use to update the Nav  position solution so that these  errors remain less than 1% distance  from the field origin and not the total distance traveled by an encoder.   This error translates to about .6 deg of heading error from a gyro update.    So we should not expect much better than that from our updates.

General Case: Lines skew to the coordinate axes

To complete this note,  lets consider a more general line that is described by

c = a*x + b*y

where a,b and c are constants and again x and y are truth coordinates.   Now we have three unknowns (x,y and delta) and three equations:

x’ = x*cos(delta) + y*sin(delta)

y’ = -x*sin(delta) + y*cos(delta)

c = a*x + b*y

First we solve for x and y from the first two equations which is a simple inverse:

x = x’*cos(delta) – y’*sin(delta)

y = x’*sin(delta) + y’*cos(delta)

Make the small angle substitution gives

x = x’ – y’*delta

y = x’*delta + y’

Now substitute x and y into the line equation

c = a*(x’ – y’*delta) + b*(x’*delta + y’)

Now solve for delta

c -a*x’ – b*y’ = (b*x’ – a*y’)* delta



delta = (c -a*x’ – b*y’ )/(b*x’ – a*y’)

Check the special case above :  y = c, a = 0, b = 1

delta = (y – y’)/(x’)  : checks ok

1508 LancerBots reach world top 40 in drive skills at Robodox skills challenge

February 26, 2013

Funny thing happened on the way to qualify for worlds in the programming challenge… the robot  won the Robodox driver challenge instead with our programmer Gevorg as the driver.  Go figure.   Lancerbots brought three robots to the competition  but only one of them was working so the programming robot did double duty as both driver and programming entry.   The drive skill score was 205 which puts it in the top 40 world rankings but you need to be in the top 30 to qualify.  Being in the top 5 of the California robots qualifies it for the California State Championships on March 16th in Santa Clara near San Jose.

I helped the kids construct a solid programming robot that was very close to being qualified for worlds…except for a pesky sack that caused the robot to not complete its route. (robot video)   The robot is a very efficent spatula robot that can pick up and dump a load of 16 sacks over its back.   It is designed with encoders  , line sensors and gyro sensors that are used to provide full navigation ( two axis position and speed as well as  heading).   The line sensors are used to update the navigation solution.

The field geometry is coded by 4 way points  which are at the intersection of  the crossing white lines used for line tracking.   The waypoints  are used to automatically steer toward using a waypoint tracker algorithm that nulls cross track distance and has heading inner loop.   It makes the programming simple and straight forward and works great so long as the gyro doesn’t drift too much.   The gyro is used to generate heading as a compass would.   We know it is not a great sensor to use as a compass because it does drift at about 6 deg/min which is enough to mess you up on the last legs of the route.   But we thought we would give it a try since it is a random phenomonon and sometimes it works great..if you are lucky.       We have some ideas to make the compass heading more reliable by using  a gyro rate deadzone and also blending the gyro with additional encoder data.

I think the robot is capable of competing at the world class level and probably could score 250 points (pending compass problem fixes and optimization).   The world record currently stands at 185 points.   Had it completed its route on Saturday, the score would have matched the current world record.  Woulda shoulda coulda…frustrating to put in hours of work and have a darn sack trip you up.

I will publish the software at a future date.   It uses four tasks: Navigation, System Control, Feedback Control and Task Control.   The User and Autonomous functions generate engage commands for automatic functions and provide  relavent parameters.

As an example:   Suppose we want to track a line crossing through a waypoint.   We simply set the waypoint number (a waypoint has a position and heading) , engage the tracker and use a “move until” distance function that stops the robot when it has reached the desired point along the track.   The lateral tracking is done automatically.   Any virtual line on the field can be tracked independent of the white lines given to the programmer to use.

Relavent posts:

Note: Using a line sensor to update gyro heading

Waypoint steering geometry for a mobile robot

Navigation update equations for a two wheeled robot

Waypoint steering geometry for a mobile robot

February 5, 2013

waypoint stearing dia

Fig 1 shows the geometry that I will use in the waypoint steering algorithms that will be discussed in future posts.  The field geometry is North (N) along the y-axis and East (E) along the x-axis.   The heading angle , psi is defined positive clockwise from North as a compass rose.

Waypoint:   The waypoint is defined with both location (X_p, Y_p) and direction psi_p.   The slope of the center track line m is  1/tan(psi_p).

Cross track and Along track Ranges:

The two parameters of interest are the along track range R_y and cross track range , R_x.   The steering is determined by R_x = R*sin(delta_psi) where delta_psi is the angle between a line drawn from the robot to the waypoint and the actual track line.

Derivation of R_x  and  R_y without sin and cos functions:

We know that the track can be expressed as a line that goes through (X_p,Y_p) and has a slope m.   So we can write the track line as  y= mx + y_0 form but we need y_0.    Plugging in for point p gives  y_0  = Y_p – m*X_p .

Hence the line eq  is   0 =  -y +m*x + Y_p – m*X_p.

It can be shown that the distance R_x  between a point  (x_r, y_r)  and a line  a*x + b*y  + c  = 0  is

|R_x|= | (a*x_r+b*y_r +c)|/sqrt(a^2 + b^2)    (see  wikipedia)

where a = m ,  b= -1 ,  c= Y_p – m*X_p ,    m  = 1/tan(psi_p)

This form can lead to division by zero when psi_p = 0 but can be avoided by using the tan(90 – psi_p)  = 1/ m  .  We  divide the starting equation

0 =  -y +m*x + Y_p – m*X_p

by m   to change its form and use it whenever m >1 or psi_p > 45 degrees.

0 = -y/m + x + Y_p/m – X_p.

Now we have new definitions of a,b,c

a = 1, b = -1 /m, c = Y_p/m – X_p  , 1/m = tan(90 – psi_p).

R_y derivation

We can use a similar method to find R_y.

|R_y| =  |(a’*x_r+b’*y_r +c’)|/sqrt(a’^2 + b’^2) )

where we use the perpendicular line to the track passing  through (X_r,Y_r) with slope  m’ =-1/m and the point as (X_p,Y_p).    This leads to

a’ = -1  , b’ = m’ , c’ = Y_r  – m’*X_r  , m’ = -1/m = -tan(psi_p)  .  We use this when psi_p  < 45 degrees and the modified form

a’ = m , b’ = 1 , c’ = -Y_r*m – X_r    for psi_p > 45 degrees.

Making R_x and R_y signed values

By convention, I want displacement to the right as positive R_x.  (X_r,Y_r) will be to the right of the track if  0< delta_psi  < 180 otherwise we change the sign of R_x.

Similarly, we want the along track distance, R_y,  to be positive if we are heading in the direction of the waypoint along the track and we have not reached the waypoint. This occurs when  0<90 – delta_psi < 180 otherwise R_y is negative.

smart_PTC_monitor beta code release

August 31, 2012

Ok, I’ve got the RobotC code used in my smart_PTC_monitor developed to a point where I would let others play with it.   I originally used a single timed iterative loop that ran in the same task as  autonomous and user modes.  I decided to create a separate task that simplifies the user interface so normal programming can occur without any of my constraints.  When done, the user functions are easily inserted into my template to seamlessly add the PTC monitor protection capability.    The program is self documented with liberal commenting but I am sure people will have a lot of questions.   I am putting it out without fully testing all the functions to get early user feedback and if it proves to have utility, I will put out a more mature version that includes user feedback.

Please post your comments to the Vex forum thread where the program is introduced.   Thanks




Below is a copy of the introduction comments in the program:

//*************************************************************************************************************** // smart_PTC_monitor_template.beta.8.27.2012.c   Rev 0

// This program is an open source beta version of a competition template for Sack Attack.  It adds a task // called smart_PTC_monitor which uses current and PTC temperature models to manage a current_limiter function // which can keep hardware PTC fuses in the motors,cortex and power expander from ever tripping.

// Program Author: Chris Siegert aka Vamfun on the Vex forums. // see my blog:  for model details and vex forum threads on subject // email: // Mentor for team 599 Robodox and team 1508 Lancer Bots

// Teams are authorized to freely use this program , however, it is requested that improvements or additions be // shared with the Vex community via the vex forum.  Please acknowledge my work when appropriate.  Thanks

// Scope and limitations: // Program assumes that servo currents are zero.  If servo position is known, then currents can be modeled. // Testing has been limited to 393 motors and 3wire motors, however, the program handles  269 motors. // Periodic updates to the models will be made and posted on the Vex forum as more testing is done. // Program handles the Power Expander PTC when the ports are entered with provided #defines // All other motor ports are automatically configured to calculate the correct currents for each PTC

// Basic Instructions: // Write your program as you normally would.  When done, put your autonomous code in place of my_autonomous() // and your user code in place of my_user().  Then do a find and replace to change all the “motor” functions to // “motor_input” i.e.  motor[Left_motor] –> motor_input[Left_motor].

// Put all your encoder speed calculations into  void compute_speeds() function and assign speeds to M[port].speed variable. // You can read these speeds from the global array whenever you need them.  This loop runs at about 100ms update rate.

// Use a #define or switch to create a // PTC_monitor_engage boolean and set true to engage the monitor.

// Tell the program which ports are used by the Power Expander as shown in the example below: // #define PE_CH1  UNUSED // if used put port number …. i.e. PE_CH1  port1  else put the #define UNUSED

// #define PE_CH2  Right_drive_motor  //use setup names

// #define PE_CH3  port3  //use port name

// #define PE_CH4  1  // use integer = to port number -1.  This is motor port2

// Initialize the PTC temperatures using these #defines: // #define T_0_DEG_F  (72.) //Set robot ambient temperature here.  If you are in a hot gym…this might be 85 deg

// #define T_M  (100.) //100 deg C =Setpoint temperature for the PTC current monitor.

// Motor currents and PTC temperature data are held in a global state matrix M[] which is a structured array that contains // the motor PTC/current data in the first 10 elements (index 0 to 9) and the non motor PTC’s data in the next three elements. // M[10] , M[11] and M[12] pertain to the cortex1_5,cortex6_10 and power expander PTC states respectively.

// Program uses two update rates: The current_limiter updates at PTC_TASK_DELAY_MS which is 15ms // and the temperature calculations are set to run at about 6 times {PTC_TASK_DELAY_MS + subtasks delays) ~= 100ms. //****************************************************************************************************************

Smart software monitor keeps PTC fuses from tripping

August 18, 2012

They say “knowledge is power”.  Well if you give a RobotC program a little extra knowledge about the state of the currents in the motors and the constants that convert currents into PTC temperatures you can make a very smart monitor.  Smart meaning, PTC’s are prevented from ever tripping while maintaining near optimal current capability.

Monitor tutorial and preliminary tests are described in my latest video(click to watch).

Keeping PTC’s from tripping by limiting current is not a new concept and both teams that I have worked with have used a simple version of a monitor that usually operates during Autonomous phase of a match where the robot might accidentally run into trouble and stall motors.   If the stall tripped a PTC, the robot would remain idle for much if not all of the driver segment of the match.  We have used a simple compare on the drive train encoder speed to shut off the move functions when the measured speed did not match a expected speed for some predetermined time.    This works pretty good for that short controlled scenario.   This type of simple monitor doesn’t work too well during driver mode since the speeds vs command don’t reflect the actual temperature state of the PTC which is the only parameter that determines whether or not it trips.    If the PTC has heated up partially an is near tripping, speed cannot be used to effectively sense this so my teams usually inhibit the simple monitors during the driver phase.

With the “noble goal” of making  monitor that “knows the state of the PTC” , I and others have been developing better models of the Vex current (given command and speed) and then using this current in a differential equation that models the thermodynamic response of a PTC to a current input.   These models have been discussed in the Vex forums and are at the point that they can be used to improve existing PTC trip protection software.  I have incorporated these models into a Smart PTC current limiter which  controls a current limiting function that is inserted between the user command and the standard motor[port] function.

The smart monitor performs the following functions:

1) reads the RobotC configuration matrix  and allocates the proper current models based upon the motors being used to feed the 1o motor  and 3 CORTEX/POWER EXPANDER PTC models.

2) initializes the safe currents for each of the motors and for each of the higher level CORTEX/PE PTC’s based upon #defines constants.

3)PTC temperatures are computed and compared against a monitor threshold temperature T_m

4) PTC Temperature Limiting:  The monitor acts as a PTC thermostat  which has a reference setting called T_m.  Under normal operating conditions, the PTC monitor logic is false and the current is limited to the maximum possible.  If the PTC models sense that the temperature T has exceeded the set point T_m, (which is slightly below the critical temperature of the hardware PTC) ,then the monitor logic trips and the current is limited to a “safe_current” .  The safe_current  is just below the PTC hold current where a PTC will never trip.  The PTC temperature stabilizes and the motor runs with limited current until the estimated PTC temperature drops to about 80% of the critical temperature.  At this level, the PTC monitor trip logic is reset and full current capability is restored.

Models used in the monitor are described a some of my recent posts.




RobotC  beta code here.

Derivation of formulas to estimate H bridge controller current (Vex, JAGUAR,Victor)

July 21, 2012

I Introduction:

In 2009 I spent a lot of time in a First Chief Delphi forum thread  with some motor experts figuring out why a JAGUAR controller was linear in speed with duty cycle and a Victor was non_linear.   An example of this phenomenon is shown here for a Vex 269 motor being driven by several types of controllers.   Data is from my Excel H_bridge model All controllers have basically the same design except for the PWM frequency.  The extremes are for the JAGUAR is 15k hz and the Victor 884 is 120 hz.

Turns out that the motor time constant tau = L/R (motor inductance/motor resistance)  relative to the PWM pulse period T is key to the phenomenon.  I call the ratio of these lamda = T/(L/R).

I developed some analytical models for this forum which can predict the waveforms and average currents with good accuracy for various types of controllers and I want to show the basic mathematical derivation in this post.   I have updated the equations to include the Vex controller which has a PWM frequency near 1250 hz which is in between the JAGUAR and the Victor 884.

It is assumed that the reader has a basic knowledge of how an H bridge works.  If not, there are many good references on the internet and I might suggest the following:

 Hbridge  Secrets

H bridge Wikipedia

The type of H bridge that I am assuming is often called a high side switcher  or  asynchronous sign magnitude H bridge.   Basically, for a given motor direction, two switches are used while the others remain open.

General note.. clicking on any figure will expand it to maximum size.

On the high side switcher, a lower switch  is closed and the upper switch  is driven by a PWM waveform and opens and closes with period T and duty cycle D.

When the PWM pulse is high, the ON current flows from the battery  through the motor as shown.

When the PWM pulse is low, the OFF phase current flows through the diode as shown.

II  H bridge Equivalent Circuit

The equivalent circuit for one motor direction of the H bridge is just a switch and a battery in series with system resistance R_s and a motor (This is modeled as an  inductor , resistor R , voltage source = V_bemf in series) as shown in the “Simplified DC motor switching model.”

There is also a diode that is across the motor to allow current to flow when the switch is opened. The switch opens and closes with a PWM drive with duty cycle D and  period T.

(click figure to expand)When the switch is closed current increases exponentially from its last value toward a steady state value of i_ss_on.   When the switch is opened, current decays through the diode toward i_ss_off.  If it reaches zero then current stops and remains at zero until the next on pulse.


We can write the differential equations for these two phases.   It is important to note that these equations are for the controller direction that puts a positive voltage across the motor.   When the motor is commanded in the opposite direction some sign changes must be made to the equations.   Where ever you see a V_b and a V_d  simply change the signs and the equations will be correct for the current.  The steady state speeds also require replacing i_free with -i_free.    These will be added in the software implementation by using the sign(command) which tests the sign of the command.   Duty is always positive and for Vex is defined as D =  abs(command/127) where the command ranges between -127 and + 127.

III ON phase Current

When the switch is closed, the voltage across the motor V_m = -V_d > 0 so the diode is reversed biased and i_d = 0 .  Using Ohms law  and the fact that i = i_L  ( no current is flowing through the diode in this phase) gives

0 = V_b – V_bemf – V_L   – i*(R +R_s).

But the voltage in the inductor V_L = L*di/dt    So we have a differential equation

di/dt = ( V_b – V_bemf – i*(R +R_s) )/L

In the steady state, di/dt–>0  so the steady state current

i_ss_on = (V_b – V_bemf)/(R +Rs)

Putting it in terms of motor time constant

di/dt = (i_ss_on – i)/tau_on   where tau_on = L/(R_s + R) .

The ON closed form solution of the differential equation can be shown to be

i_on =  i_0*exp(-t/tau_on)  + i_ss_0n *(1 – exp(-t/tau_on)) 

where  i_0 is the initial current.

At t = DT  the maximum occurs and using the substitution  lamda = T/tau_on  we can write

i_max = i_o*exp(-lamda*D) + i_ss_on*(1 – exp(-lamda*D))

IV OFF phase current

When the switch is opened current in the motor coil finds a path through the diode.

i_L = i_diode

0 = V_bemf + V_L + i_L*R  + V_d

Substituting for V_L = L*di_L/dt  gives the differential equation

di_L/dt =- (V_bemf + V_d  + i_L*R)/L        and

i_ss_off = -(V_d + Vbemf)/R             when di_L/dt –> o.

The OFF closed form solution is

i_off =  i_max* exp(- t’/tau_off) +  i_ss_off *(1 – exp(- t’/tau_off))

where again tau_off = L/R .    Here i_max is the maximum current of the ON phase which occurs at the exact point that the switch opens or at t = D*T so   i=max is the initial current for the OFF phase.    t’ is the time from the switch opening.   I. E.   t’ = t – D*T  for t> DT.

V Approximation for Vex motors

The  time constants  for the ON and OFF  phases are slightly different due to the addition of Rs during the switch ON phase.    I.e.  tau_on = L/(R+Rs)   and  tau_off = L/R.   For the Vex motors, Rs << R so the following equations will use the approximation  tau= L/R = tau_on =tau_ off.

VI Final Current Value,  i_final

During the OFF phase the current will decay to a final value of current.  The point  in time where it reaches it final value will be defined  t'(final) = D’T .   Substituting t’ = D’T  and lamda = T/tau  into i_off  equation gives

i_final  =  i_max*exp(- lamda*D’) + i_ss_off*(1 – exp(-lamda*D’)

VII Initial current i_0

A steady state solution requires that

i_0 = i_final

otherwise after a cycle is completed  the next initial condition will be different from the initial condition for the last cycle.

Using this equality and substituting i_max into the i_final equation gives

i_0 =( i_o*exp(-lamda*D) + i_ss_on*(1 – exp(-lamda*D))*exp(-lamda*D’) + i_ss_off*(1 – exp(-lamda*D’)

Solving for i_0 gives

i_0 = ( i_ss_on*(1 – exp(-lamda*D))*exp (-lamda*D’) + i_ss_off*(1 – exp(-lamda*D’))/(1 – exp(-lamda*D)*exp(-lamda*D’))

(Click to expand figures)

Recall as shown in this figure, the current can be continuous or discontinuous.

Case A: Continuous current

If the diode conducts during the whole OFF phase then steady state current requires i_0 = i_final and

D’ = 1-D .  That is i_final occurs exactly at the end of the period.  Substituting for D’ and i_final in the i_o equation gives

i_0 = ( i_ss_on*(1 – exp(-lamda*D))*exp (-lamda*(1-D))+ i_ss_off*(1 – exp(-lamda*1-D)))/(1 – exp(-lamda*D)*exp(-lamda*(1-D)))

Case B: Discontinuous current

In Case B, the current tries to reverse sign during the OFF phase but is clamped to zero by the diode for the remainder of the PWM period.   So both the final and initial currents are zero in the steady state.

i_final = i_0  = 0

With this constraint, we can solve the i_0 equation for D’ at the point where the waveform reaches 0.

0 =  i_ss_on*(1 – exp(-lamda*D))*exp (-lamda*D’) + i_ss_off*(1 – exp(-lamda*D’)

exp(-lamda*D’) =  -i_ss_off/(i_ss_0n*((1-exp(-lamda*D) – i_ss_off))

and taking the ln of both sides

D’ =  – ln( -i_ss_off/(i_ss_on*((1 – exp(-lamda*D) – i_ss_off)* lamda)

Or in terms of i_max,

 D’    =  – ln( -i_ss_off/(i_max – i_ss_off)* lamda)


Our primary interest in developing current equations is to determine an expression that can be used to control the long-term behavior of a motor relative to a slow PTC fuse or setting a the steady state speed for a robot.    What we want is the equivalent DC current over a cycle  which is the average current per cycle.   This is computed by integrating the area under the current response and dividing by a cycle time.  If the times are normalized to the controller period T then the average current

i_avg = Area_on + Area_off

where Area_on and Area_off are the areas under their respective current responses in the ON and OFF phases.  By integrating the i_on with respect to t and normalizing to the period T gives

Area_on = i_ss_on*D – (i_ss_on – i_0)*(1 – exp(-lamda*D)/lamda

Similarly for the off phase we have the same form except i_o for the off phase = i_max.

Area_off = i_ss_off*D – (i_ss_off – i_max)*(1 – exp(-lamda*D’)/lamda

Believe it or not  , the second term in the Area_on equation cancels the second term in the Area_off equation.   Both these terms can be related to i_max – i_0  by using the fact that i_o = i_final.   With some substitutions and math manipulations it can be shown that:

(i_ss_on – i_0)*(1 – exp(-lamda*D)   =  -(i_ss_off – i_max)*(1 – exp(-lamda*D’) = i_max – i_o.

This leaves

i_avg = i_ss_on*D + i_ss_off*D’ 

We can substitute for i_ss_on and i_ss_off and get i_avg in terms of voltages.

i_avg = (V_b – V_bemf)*D/R  – (V_d + V_bemf )*D’/R

For a given D  , D’ can vary from 0 to 1 – D  depending upon the value of D and lamda.   The current waveforms take on essentially three shapes:  rectangular, triangular  or exponential saw tooth shapes depending on how D’ is characterized.

XI   Vamfun rectangular approximation      lamda>>1 , D’= 0.

Under this condition the motor responds very quickly with respect to the PWM pulse.  It essentially reaches I_ss_on  and stays there for the duration of the ON pulse and quickly decays to zero during the OFF phase.   Thus the current waveform approaches a rectangular shape and so we can compute i_avg accurately with this formula when D’ = 0 :

i_avg = (V_b – V_bemf)*D/R

This leads to a steady state speed vs duty that is inversely proportional to duty.   We know that the free speed of a motor occurs when i_avg = i_free.  Rearranging the current equation with this equality gives

V_bemf(steady state)  = V_b – i_free*R/D      for    D>= i_free*R/V_b

= 0     for    D < i_free*R/V_b  

which says that if the current doesn’t exceed i_free then the drag torque holds the motor speed to zero.

This figure shows a typical rectangular waveform produced by a Victor 884 controller  ( PWM frequency = 120 hz , lamda = 30) controlling a Vex 269 motor. (click to expand)

This curve was generated with a Labview program I wrote which is a simple integration of the L R response to various duty cycle inputs.

X  Vamfun triangular approximation        lamda <<1     ,  D’ = 1 – D

This approximation occurs when the PWM frequency is much higher than the motor response frequency and many PWM cycles occur in one tau for the motor.    Both the ON and OFF phase waveforms are approximated by linear slopes and the waveform is triangular.     The approximate formula becomes

i_avg = (V_b – V_bemf)*D/R  – (V_d + V_bemf )*(1-D)/R   or collecting D terms

i_avg = (V_b +V_d)*D/R  – (V_d + V_bemf)/R 

This leads to a linear steady state speed vs duty  when i_avg = i_free.   I. E.

V_bemf(steady state) =  (V_b + V_d) *D    –  i_free*R  – V_d   

for  D    >  (i_avg*R  + V_d)/(V_b  +V_d)


V_bemf = 0.

This equation  becomes  the standard DC motor current model when V_d = o .

This linear speed response would be typical of a Vex 269 being driven by a TI  JAGUAR  controller  that operates with a PWM frequency of 15 kHz which makes lamda = .24 .  The JAGUAR controller is used in First robotic competitions.  It  produces  a typical triangular waveform as shown here (click to expand figure):

XI  Non extreme lamda produces exponential saw tooth:

This figure shows an example of an exponential saw tooth waveform produced by Vex controller (PWM frequency 1250 hz , lamda = 3) driving a Vex 269 motor with a .3 duty.      (click to expand)

Notice that the first trace which is the motor voltage has three distinct levels.  During the ON phase it equals V_b.  During the off phase while the diode is conducting it equals       -V_d and when the current goes to zero it then  reads the back emf voltage V_bemf.   During this period some controllers actually sample V_bemf to estimate motor speed without encoders.  If the diode always conducts like the previous figure, then the controller must periodically use a special pulse to sample V_bemf.

XII Computational Proceedure:

Ok, we have lots of formulas and here are the steps to put them to use.

0) Gather inputs: V_b,V_bemf,V_d, lamda, R, command

1) Compute the input duty D and direction dir

D = abs(command/127)

dir = sign(command)

2) Compute  i_ss_on and i_ss_off from known voltages.

i_ss_on = (V_b*dir – V_bemf)/R

i_ss_off = -(V_d*dir + V_bemf)/R

3) Compute i_0 and test if current is continuous or discontinuous.

i_0 = ( i_ss_on*(1 – exp(-lamda*D))*exp (-lamda*(1-D))+ i_ss_off*(1 – exp(-lamda*1-D)))/(1 – exp(-lamda*D)*exp(-lamda*(1-D)))

If the sign(i_0) is opposite the sign (i_ss_on).   I.e.

if( sign(i_0) *sign(i_ss_on) < 0 ) then  current is discontinuous and  set i_o = 0.

4)  Compute i_max

i_max = i_o*exp(-lamda*D) + i_ss_on*(1 – exp(-lamda*D))

5) Compute D’

if(discontinuous) then

D’    =  – ln( -i_ss_off/(i_max – i_ss_off)* lamda)


D’ = 1-D 

6) Compute i_avg

i_avg = i_ss_on*D + i_ss_off*D’

Thats it.   See link below for a RobotC implementation of these equations.

Related Links:

Vex 269 Current data based upon Excel Hbridge Model

proposed-method-of-incorporating-ptc-resistance-into-my-vex-motor-models/RobotC subroutine to compute Vex 269 motor currents


Vex threads:

Estimating 269 Motor Current based on H bridge models

Estimating motor current


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