I am designing a new mobile robot platform and i need a generic solution for the robot in order to establish a proper mathematical modeling. My question is about calculation of the traction force of the robot with respect to rotational speed of the direct contact motor shafts which is mounted as the figure below.

Contact points of the vehicle, shafts and spherical wheel

There is a vehicle with 3 contact points with the ground. Two contact points are edges of steel cylindrical shafts with diameter d_c. Other contact is provided by spherical wheel with diameter d_s in order to provide static balance.

Related parameters are given on the diagram

Cylindrical shafts are rotating with a high speed, n. The angle between the shafts and the ground is a. Coefficient of static friction is f. Shaft is slipping on the ground because of high speed and low friction. The weight of the vehicle is w. The distances between the contact points and the center line is b.

How to calculate traction force T with respect to rotational speed n for a shaft mounted at a certain angle a on a surface?

Given Parameters:

These parameters are just for example, i need a generic mathematical solution.

d_c = 1 mm

d_s = 10 mm

b = 25 mm

n = 10000 rpm

a = 60 degrees

f = 0.25

w = 0.1 kg


T(n, a) = ?


2 Answers 2


You will have dynamic friction here, which is independent of the rpm of the rotating shaft, because it is slipping continuously. so you need to have the coefficient of dynamic friction, let's call it k.

Assuming each cylinder take 1/3 of the mass of your robot, it generates a traction of $ \ T=k*m/3 $

  • $\begingroup$ So, you say that i can't control the speed of the vehicle just by changing the DC motor power since traction does not depend on the rotational speed of the shaft. It doesn't matter the motor runs at 500 or 50000rpm. Correct? $\endgroup$
    – user21307
    Jul 31, 2019 at 19:14
  • $\begingroup$ Yes, unless you reduce the speed to below any slippage. If you do that then the speed of your robot is the same as the speed of the wheels. Because in a rolling motion between to semi rigid surfaces the friction is negligible. So if we consider the friction at the start up static friction rises to its maximum without any movement of robot till it jerks to move, then it will drop to near zero, while the speed of robot is the same as that of the wheels. Now if you push more power to start slippage the force is not going to increase no matter how fast the wheels turn. $\endgroup$
    – kamran
    Jul 31, 2019 at 19:43
  • $\begingroup$ How about the angle, a? Do i need to optimize this parameter? $\endgroup$
    – user21307
    Aug 1, 2019 at 10:28

No, not even angle. If you have seen drag racing, they have high traction expensive tires. But when before the race they burn the tires to creat sticky hot surface, while creating clouds of smoke by revving the engine the car doesn't move forward. Because they have forced the tires to slip and skid.


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