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14.1. Kinematic analysis of the McNamee wheel

14.1.1 McNamee Wheel Introduction

In the era of industrial manufacturing, people have felt that the steering of traditional vehicles is very inconvenient to use in some specific environments, for example, in the aerospace field of component assembly, the use of traditional vehicles to carry assembly will consume a lot of manpower, material resources and time. In the case of precision to a few millimeters, if the alignment is not correct, it can only be recalibrated. For highly sophisticated large-scale equipment, high efficiency means being able to outperform the competition by a few blocks. If you can find a vehicle without turning the body can be any angle of translation, rotation, can be a perfect solution to this problem.

Until 1973, the Swedish company McNamee developed a more practical program has been widely adopted, which is the McNamee wheel we will introduce today.

McNamee wheel (hereinafter referred to as McWheel) is composed of the hub and the rollers around the hub, the rollers are a kind of unpowered slave small rollers, McNamee wheel roller axis and hub axis angle is 45 degrees, and there are two kinds of mirror image of each other's A and B wheels, or it will be known as the left-handed wheel and right-handed wheel. This will generally be labeled A and B, L and R on the top of the hub.

14.1.2. McNamee Wheel Motion Analysis and Direction Determination

It was mentioned earlier that the McNamee wheel is divided into two kinds of AB, if the A wheel moves forward while moving to the right, i.e., moving diagonally to the right front, then on the contrary, the A wheel moves backward while moving to the left, i.e., moving diagonally to the left rear; accordingly, the B wheel can move diagonally to the left front and the right rear.

Taking the front of the cart as the positive direction, it is agreed that the direction when the wheel is moving forward is the forward rotation of the motor, and the direction when the wheel is moving backward is the reverse rotation of the motor. (All of the following lessons are illustrated in this direction)

Take the A wheel, the roller moving direction due to rolling and thus can not provide forward force, and in the direction of the roller axis roller can not roll and friction with the ground to produce the roller axis upward friction, that is, diagonal to the right before or after the direction of the left, and thus A wheel of the direction of the speed of the diagonal to the right before or after the left; Similarly, can analyze the B wheel.

From the physics we learned in high school, we know that velocity can be decomposed orthogonally, determining that the motion of the car depends on the direction of the combined velocity of the four wheels.

Then wheel A can be decomposed into velocity components that are axially to the right and vertically axially forward, or axially to the left and vertically axially backward.

In this way, the velocity components of the B wheel and the A wheel are now in a mirror image relationship with each other.

14.1.3. McNamee Wheel Arrangement Combination and Motion Analysis

After knowing the velocity components of the AB wheels, we can arrange and combine the McNamee wheel chassis of the four wheels: AAAA, BBBB, AABB ......

Are all combinations capable of full range of motion? Not really.

Below we will give just one of the wrong examples and one of the right ones, and the rest can be reasoned out on their own (there are actually too many).

Analyzing the wrong example [AAAA]

This one doesn't make sense at first glance, but let's analyze it anyway because as an example of a mistake, it's very easy to analyze what exactly is wrong with it. Earlier we talked about how the velocity components of the A wheels are either forward plus right or back plus left.

When all four wheels turn forward at the same time, each wheel will have a speed component to the left, which will lead to the whole chassis forward will necessarily move to the left at the same time; similarly, backward will necessarily move to the right, so there is no way to use it, this thing is uncontrolled to run around, which is not the omnidirectional movement we want.

The correct distribution of wheat whorls should be [ABBA]

When all four wheels are turned forward, the AB wheels can cancel each other's axial speed, leaving only the forward speed, so that the chassis is straight ahead and will not run away, and the same for backward;

If the forward and backward velocities cancel out when wheel A is spinning forward and wheel B is spinning backward, leaving only the velocity to the right, then the chassis will translate to the right;

Conversely, if wheel A is reversed and wheel B is forward, it will pan to the left;

Finally the chassis will rotate to the right by turning the left wheel forward and the right wheel in reverse; conversely, the chassis will rotate to the left.

Also, when the A wheel is turned positively and the B wheel is not moving, the chassis moves to the right front;

The chassis moves to the left rear when the A wheel is reversed and the B wheel is not moving.

Accordingly, the chassis moves to the left front with the A wheel not moving and the B wheel rotating positively;

If the A wheel is not moving and the B wheel is reversed, the chassis moves to the right rear.

There are also four other ways to move around the midpoint of the axis.

[Think] Aren't the above four cases very similar to translations, when the two wheels that are not moving move like a translation, but at a speed less than that of the other two wheels, how will the McNamee chassis move?

14.1.4. McNamee Wheel Mounting Positions

After we finished explaining the McNamee wheel mounting combinations earlier, there is more to be said about where to mount the four McNamee wheels.

The shapes enclosed by the contact points of the four wheels with the ground are mainly squares and rectangles. Oddball shapes are not excluded, but only squares and rectangles will be analyzed here.

Square: The four wheels are located at the four vertices of a square, and there are no problems with translation and rotation. Due to the shape and size of the robot's chassis, this type of mounting is ideal but unattainable.

The chassis we use with the addition of the Wheat Wheel is a carefully designed square, making it the ideal Wheat Wheel chassis.

Rectangular: Wheel rotation produces a Yaw axis torque, and the force arm of the torque is longer. It is the most common type of mounting.

By comparing the traditional vehicles, we can find that the wheels of ordinary vehicles are turning in one direction when moving, while the car using wheat wheels is doing omni-directional movement, as analyzed above, the direction of movement of each wheat wheel is different, so the wheat wheels want to do a real omni-directional movement, each wheat wheel needs a separate motor to drive, in this way, a set of control system is also needed to grasp the Each wheel's steering and rotation speed.

To realize omni-directional movement generally use omni-directional wheels and mcnemonic wheels, above we mainly introduced the mcwheel movement and installation combinations. Omni-directional wheels and McNamee wheels have differences in structure, mechanical properties, and kinematic properties, which is essentially due to the difference in the angle between the wheel hub axis and the roller rotating axis. After analyzing, the difference between the kinematic and mechanical characteristics of the two can be reflected by the following table.

The calculations are shown below for reference only:

14.1.5 Final installation method