# What torques or acceleration is required to move a differential drive robot on a circular path?

When a differential drive robot moves on a circular arc with constant speed, its kinematics gets me constant wheel velocities. This appears to suggest that once a circular motion is reached and curvature remains constant, no acceleration and hence no motor torques are required. It also fits my understanding that circular motion at constant speed requires no work, as the tangential displacement is orthogonal to the radial centripetal force.

Intuitively however, I cannot wrap my head around this - it seems to not match how such systems actually behave - the centripetal acceleration experienced very much limits how much it can accelerate tangentially. Furthermore I find it hard to believe that change in momentum achieved by changing the direction of velocity should come “for free”.

I only have a superficial understanding of mechanics, so I’m sure there are a lot of holes in my thought process and I would be grateful for someone to point them out.

It doesn't come for free, it involves one of the most important forces that not only allows wheeled robots and other vehicles to turn but also enables people to walk! Namely, Friction :)

In an ideal situation, where the vehicle is maintaining constant speed, the total work done on the wheels is indeed zero (cf. Pure Rolling), because the entirety of the frictional force is converted into the kinetic energy of the wheel's rotation. However if, as you mentioned, keep on accelerating while trying to maintain the vehicle on the same radius $$R$$, the centripetal force $$F_c$$ required will grow proportionally to the instantaneous speed due to $$F_c=\frac{mv^2}{R}$$ until the point in which the surface will no longer be able to deliver the necessary frictional force, and the vehicle will slide.

When sliding, work is definitely done on the wheel because the sliding of a wheel just means that the same point on the wheel remains in contact with the surface for some $$\Delta{t}\gt 0$$, so that the frictional force is doing work on that point (and many points in general) proportional to the displacement $$\Delta{x}$$ traversed during that time interval.

In practically all mechanics courses, students are asked to calculate the necessary friction coefficient required to maintain a car on a turn of a specified radius and with a specified speed, you can find a nice example of that here. I also recommend taking a look at the very intuitive answer given by @Wouter to the question: Why does friction cause a car to turn?

• Thank you, those are some good points, but they don't complete the picture for me yet. Friction is required to make circular motion possible on a plane normal to gravity, but how do I incorporate friction into the kinematics or dynamics of my robot? How does the existence of friction impact the acceleration required for linear and circular motion? Is there a model that incorporates friction required for circular motion while idealizing other aspects like slip? Commented May 29, 2023 at 22:30
• @MonkeyKhan Well, what you have control over are the robot's wheels or tyres (I don't know what material they're made of) and to a lesser degree probably, you may have some control on the types of surfaces on which you allow the robot to move. Tyres with bigger surface area will make it easier for a frictional force to develop between them and the surface. Depending on how fast this machine will be moving, you may have to consider also Tyre treading but that may be an overkill for low speeds. Anyway, this question is somewhat beyond what you originally asked about, isn't it? :)
– Amit
Commented May 29, 2023 at 22:39
• @MonkeyKhan Also allow me to point out, that there is a Robotics.SE, so if you find that you have a question that is very much focused on the specifics of how to implement this mechanism in a robot, it may be better suited there.
– Amit
Commented May 29, 2023 at 22:52
• > Anyway, this question is somewhat beyond what you originally asked about, isn't it? :) Well no, I did ask about torques or acceleration required to achieve circular motion. You rightly pointed out that my original line of thought disregarded friction, but I would still like to know what incorporating friction means for acceleration required. As for robotics - I could also have gone with a car, like most of examples do, but differential drive is as simple as it gets if you want to talk about actual wheel torques rather than just some accelerated mass. Commented May 29, 2023 at 22:59
• @MonkeyKhan Yes, you will need to implement some kind of differential. The open differential is a very common option. Another thing you usually need to take care of is what's known as "Ackermann steering"... that's more a matter of geometry however. See if the wheels are connected by a straight axle not only are they turning on different radii but the outer one is also lagging behind the inner one's radius (see diagram), so it has to make a different angle with the road...
– Amit
Commented May 29, 2023 at 23:30