R2B4 #3 - Drive Motor Calculations, Drive Train Design and Control Design

1.     Overview

I plan on using NEMA17 stepper motors to drive R2B4 along. They appear to be easy to control and provide a lot of torque...but will it be enough? What size wheels do I need and how quickly do I need R2B4 to move? How fast can I step the motor? Do I need any extra gears? These are my design calculations to try an make something that can actually move...then I'll see in reality. I'm limiting my total vehicle weight, including the load of sand, to 10Kg regardless of the load cell, power source and chassis weights; that will make it fairly portable. I initially opted for a nominal wheel diameter of 89mm as it looks good and I was sure I had a holesaw that I can use to cut the wheels with from plywood.....after all the calculations I found that wasn't true. Instead I used my largest holesaw to cut some 70mm diameter plywood disks for the wheels; they will likely also get some tread so will be bigger diameters afterwards.

2.     Motor Traction

2.1     Calculation of Motor Traction

The ground is likely to be covered in old sand/gravel and the chances of getting both wheels to provide traction is slim at times. Therefore I will work towards a worst case that only one wheel will provide the required traction to get the R2B4 moving. My calculations are not the most interesting things to all and when I drafted this blog it was really dull to look at. So I'm wrapping them up into a tabbed section, those that are interested can delve in, everyone else can skip over

{tabbedtable} Tab Label	Tab Content
The Rolling Resistance	Rolling Resistance is the required force to overcome the friction between R2B4 and the ground. I'm going to take the ground as a fairly bad case of medium quality concrete (surface friction constant = 0.015).       Rolling Resistance          = Vehicle Weight (Kg) X Surface Friction Coefficient                                         =     10 x 0.015                                         =      0.15 Kg
The Gradient Resistance	I'll need a greater motor tractive force to move R2B4 up a slope. Luckily most builders mix their sand and cement on a fairly flat area....I'm assuming nothing greater than 1:20 gradient. This equates to 2.86 degrees or rounded down by me (because I can set the limit to whatever I like) to 2.5 degrees.       Gradient Resistance                 =     Vehicle Weight (Kg)     x     Sin (slope angle)                                                  =          10 x      sin (2.5 degrees)                                                  =     0.44 Kg
The Acceleration Force	Remember that Force = Mass multiplied by acceleration? So I'll need a force to also get my R2B4 moving, the size however depends on how quick I want it to get to full speed and what that speed is. Again, I can set values so I'm thinking 2 seconds to reach full speed and a full speed of 0.5m per second would be ample (actually it might be too fast....but is the most taxing case).        Acceleration Force     =     Weight (Kg)  x  Maximum Speed (m per second)  /  [  Acceleration due to gravity(m per second per second)   x Time to Reach Full Speed (seconds) ]                                         =     10  x  0.5 / (9.8  x  2)                                         = 0.255 Kg
Interlude (grab a coffee or have a nap )
Adding It All Up	The total tractive effort required by the motor (for one in my case) is the sum of the three sections before.     Total Motor Traction      = 0.15 + 0.44 + 0.255                                      =     0.845 Kg
Torque Required by Motor	The motor torque required (r=4.45cm)            =          Total Motor Traction  x   Radius of Wheel (cm)    x  Resistance Losses                                                                         = 0.845   x  4.45    x     1.15 (nominal)                                                                         = 4.32 Kg.cm                                                                         = 42.4 N.cm   The motor torque required (r=3.5cm)              =          Total Motor Traction  x   Radius of Wheel (cm)    x  Resistance Losses                                                                         = 0.845   x  3.5    x     1.15 (nominal)                                                                         = 3.40 Kg.cm                                                                         = 33.33 N.cm   The resistance losses are to take account of the bearing friction, this value seems about reasonable from looking around the internet although could be worse once the grit gets in there!
Wheel Spin	I'm not thinking De Lorian (as in Back to the Future), but if my motor torque is too great and the friction between the wheel and ground is too little and/or the load is light, then R2B4 will loose traction. Taking a poor case of hard plastic wheels on a concrete surface I would have a Static Coefficient of Friction of 0.3     The maximum tractive force that can be applied with r=4.45cm (N.cm)  = Weight (Kg)  x  Static Friction Coefficient   x   Radius of Wheel (cm)    x   9.8 (m/s/s)                                                                                                               =    10   x   0.3    x   4.5    x  9.8                                                                                                               =  132 N.cm  (for r=4.45cm)   The maximum tractive force that can be applied with r=3.5cm (N.cm)  = Weight (Kg)  x  Static Friction Coefficient   x   Radius of Wheel (cm)    x   9.8 (m/s/s)                                                                                                              =    10   x   0.3    x   3.5    x  9.8                                                                                                              =  102.9 N.cm  (for r=3.50cm)
Useful Reading	http://engineershandbook.com/Tables/frictioncoefficients.htm http://www2.mae.ufl.edu/designlab/Class%20Projects/Background%20Information/Friction%20coefficients.htm http://www2.mae.ufl.edu/designlab/motors/EML2322L%20Drive%20Wheel%20Motor%20Torque%20Calculations.pdf

2.2     Traction Summary

My NEMA-17 has a part number of 17HB19-2004S1 and is manufactured by OSM. From their website datasheet I can see that this can produce 59 N.cm and therefore if both my calculations are correct, even if one wheel looses traction, the other should suffice to get it going again (within the limits of weight, incline and ground resistance)....we'll see if that is true shortly. Also being able to power the R2B4 via a single motor will enable me to steer it (I think?) although that statement could be incorrect as I'll likely power the other motor in an opposite direction.

It would therefore seem unlikely that the wheels will slip when turning at the maximum torque of 59 N.cm (at 2A current ). I have also undertaken the calculations that the vehicle load is over each wheel....maybe I should have used a quarter of that value? Another point would be that the load will not always be evenly distributed over the R2B4 and could actually lighten the load on the drive wheels bringing the Maximum Tractive Force that can be applied tumbling downwards and causing wheel slippage.

The smaller the diameter of the wheel the greater chance that, for the same applied torque, that the wheel will loose grip. So my 89mm diameter wheels would have been nice but 70mm diameter will suffice. I can always add some beading/tread to increase the diameter and the grip.

3.     Speed Calculations

I'll need to drive the stepper motors with pulses to get the desired maximum speed, based on the wheel diameter and the acceleration limits I placed on it. I can do that in software gradually increasing the speed to give an acceleration envelope/profile. I have a 1.8 degree step angle on this NEMA-17, which equates to 200 steps for one revolution.

For my maximum speed of 0.5m/s (which I still think might be too fast) I would need the following number of wheel revolutions:

     Wheel Revolutions     =     0.5 (m) / (2 x PI x r)

                                        =   2.274

     I need 200 pulses per revolution

     Therefore I need 454.8 pulses per second (at top speed)

4.     Hardware

I bought a pair of L298 stepper motor H-bridge drivers for this project not realising the dual row of pins are not aligned to 0.1" pitch  - so I cheated and ordered up some pre-made boards using L298 chips:

The good news for me is that these can drive up to 2A, and my NEMA-17 is also happy to operate at 2A (if fact that is where I'll get my 59N.cm torque at) so hopefully there is nothing I can damage if the adjustment potentiometer is incorrectly set.

These are going to be driven directly from the SensorTile Cradle, although I will do a few current tests just to check what the unit draws. A useful diagram and description can be found here. The power supply to this board for the NEMA-17 is going to be a Sealed Lead Acid battery, in fact here it is:

But there is something I didn't really consider, and that is how heavy the battery is. It weights about 2.5Kg - that is a considerable percentage when my calculations are based on a total weight of 10Kg. Anyway I'm pushing on and will see what happens.

5.     Drive Path

To avoid damaging the internal bearings of the NEMA-17 by crushing them under the axial loading I need to take the main R2B4 weight onto some separate bearings. The drive shafts will be 8mm diameter steel rod, that are to be threaded on the ends to make the wheels captive (via a retaining nut and washer). They will pass through the additional bearing block. The part of the axle that protrudes will get coupled to the NEMA-17 via a semi-flexible metal coupling (that I'b be making tomorrow hopefully).

This will determine the width of R2B4 starting with the two back-to-back NEMA-17, the couplings and the width I decide to make the external bearing blocks to. I have thought about offsetting the two motors so they take up the same width on the centerline of R2B4...but that might be really hard to steer?

6.     Making Wheels

Today I was trying to make some wheels starting with the 89mm ones. I was convinced I had something from the old BBQ that was about that size but didn't manage to find them. I then started looking at a nice chunk of Ash tree - almost round and quite straight. I could get some decent wheels from that perhaps?

In the end I cut out some plywood squares 100 x 100mm, rounded the corners, drilled the centre to 8mm and stacked them onto a threaded bar. The idea was going to be to lathe them down to 89mm. But I stopped for a couple of reasons:

•      I don't actually have a wood lathe - just a wooded thing I made that takes my electric drill.
•      I needed to use the wood router base to hold the drill - then realised it was still loaned out to someone
•      I did the calculations using my largest holesaw of 70mm and found that might work
•      I realised 70mm holesaws produce a lot of friction in plywood - I felt like I was a caveman making fire for the first time
•      Lastly, I know as the grain of plywood runs in various directions, and I had laminated them, it would likely result in large chunks getting ripped out (and perhaps from my hand)

I also thought about casting them from car body filler, hot melt glue or some sort of sawdust and glue amalgam. In the afternoon I went to the shops and scoured the toy aisles for something that I could use the wheels from but nothing fitted the requirement. There is a local carboot sale tomorrow that would work well.

7.     What Next

Now I have planned my approach out, and I am optimistic that R2B4 might actually move, but I need to get making, I must also remember to take my camera out to the shed to get some photos along the way. Work to do includes:

• Cut some wheels from plywood
• Make bearing blocks and drive shafts
• Make drive shaft couplings
• Get the SensorTile Cradle Extension setup for pulses at 455/second max and perhaps work a function to ramp that up linearly
• Use the SensorTile Cradle Extension to drive my steppers via the L298 module

As I may have already said, I've never made something like this before, and if there is a glaring obvious mistake then please let me know.

I hope that my blog#3 is interesting to anyone else making a bot or similar.

Rod