A wheel is one of the best ways to move a robot, but only on some ground. On a smooth floor a robot rolls far on a small battery. On sand the same robot stops almost at once. The reason is that wheels and ground squash a little, and squashing costs energy. This page follows that idea step by step with a simulator at every step: a race on four floors, a perfect wheel, a squashing wheel, a wheel made of springs, spring energy graphs, rolling friction and a wheel in sand. Everything here is a simulation, so no robot is needed.
Wheeled movement is very efficient on some ground and very inefficient on other ground. Hard and flat ground, like a smooth floor, asphalt or concrete, is good. Soft or uneven ground, like sand, mud, grass or loose soil, is bad. Here is the same robot with the same battery on four different floors. Press the start button and watch which one gets furthest.
To see what goes wrong in real life, first look at a perfect case. Take a rigid wheel, which never squashes, on rigid ground, which never squashes either. A steel train wheel on a steel rail is a good example. Because nothing squashes, the wheel and the ground touch at one single point, called P. A force F pushes the wheel, and the wheel turns with angular velocity ω. Now imagine this happening in a perfect vacuum with no energy loss at all. After one push, the wheel would roll for ever.
In the real world, wheels and ground are not perfectly rigid. At the place where they touch, the wheel squashes and often the ground does too. So the contact is no longer one point. It is a patch with a size, called the contact surface S. How much squashing happens depends on three things: the weight of the vehicle, the material of the wheel (softer rubber squashes more than steel) and the material of the ground (soft or uneven ground squashes more). A steel wheel on a steel rail squashes very little. A rubber wheel on a heavily loaded truck squashes much more. Even when the squashing is very small, it still exists.
A helpful way to think about the squashing is to imagine the wheel is made of many small springs. When one part of the wheel reaches the ground, its springs are squeezed. This is compression. As the wheel turns, that part leaves the ground, and its springs spring back out to normal. This is decompression. So every bit of the wheel goes through the same cycle again and again: compression, then decompression.
To understand what happens in the wheel, first look at one simple spring. Push on it with a force F and it squeezes by a distance x. The bigger the force, the bigger the squeeze, in a straight-line way: F = k × x. This is Hooke’s law. If the force and the squeeze are drawn as a graph, the area under the line is the work (energy) used to squeeze the spring. That energy is stored in the spring. When the force is removed, the spring springs back and gives the energy back. For an ideal spring, the energy put in equals the energy recovered, so an ideal spring has no energy loss.
The real world is different. A real spring does not give back all the energy that went in. Squeezing and letting go follow two different lines on the graph. At first a small force squeezes the spring, and later much more force is needed for a little extra squeeze. On the way back out, the spring pushes with less force than it took to squeeze it. The area under the pressing line is the work needed to squeeze the spring. The area between the two lines is the energy that was lost, mostly as heat inside the spring, and sometimes as a lasting change of shape. So for a real spring, the energy used to compress it is more than the energy recovered.
A real wheel is like a real spring. Every turn, the part at the ground is squeezed and then springs back, and not all of the energy comes back. The energy lost in this squeezing and springing back is called rolling friction. The bigger the energy loss, the bigger the rolling friction, and the lower the efficiency. So the key chain is: more deformation, more energy loss, more rolling friction, lower efficiency. The boxes below follow that chain with numbers. They are teaching values, close to real ones for common materials.
Now take a wheel on soft ground like sand. Here both the wheel and the ground squash, so both of them lose energy. The wheel is still a set of springs that squeeze and spring back. The ground can be pictured as many small springs too, with one big difference. A wheel spring goes compression, then decompression. A sand spring goes compression, then permanent deformation. The sand does not return to its old shape, which is why a wheel leaves tracks. The energy used to squash the sand is not recovered. It goes into making a lasting dent in the ground.
Here is the final explanation in one picture. On hard ground, only the wheel squashes a little, so some energy is lost and efficiency is relatively high. On soft ground, the wheel squashes and the ground squashes too. That is a much bigger loss, and because the ground stays dented, a large amount of energy ends up inside the ground. Change the wheel and the weight to see how the two parts of the loss change.