Welcome to ROSMASTER R2 repository

14.2 Kinematic analysis of the Ackermann trolley

14.2.1 Introduction of Ackermann models

Features of the Ackerman car structure: the car has 4 wheels in total, the front 2 wheels are servo steering wheels, and the back 2 wheels are driving wheels, which are mainly responsible for the overall movement. The difference with other kinds of cars is that it has its own unique servo steering, while other kinds of cars are adjusted to turn by the difference in speed between the left and right wheels. The main parameters of the Ackermann model include left and right wheelbase, front and rear wheel spacing, and turning radius.

Conventional automobiles use mechanical differentials to accomplish differential speeds, and the basic law of motion of a mechanical differential is that the sum of the rotational speeds of the driving wheels on both sides is always equal to two times the rotational speed of the differential case, regardless of whether the vehicle is turning or going straight. For example, if the rotational speed of the differential case is V, the rotational speed of the inner rear wheel is V-a, and the rotational speed of the outer rear wheel is V+a.

In recent years electric vehicle companies have risen rapidly, and in electric vehicles, the use of two motors to directly drive the rear wheels eliminates the need for a mechanical differential, making the mechanical design simpler. This makes it a little more complex in terms of electrical control. The front two wheels of the chassis are connected by a linkage that is used to control the direction in which the wheels run. A servo is usually used to drive the front two wheels to control the direction of the robot.

14.2.2. Ackermann structural dynamics modeling

Ackermann model steering characteristics:

• When driving before, the axes of the 4 wheels are parallel to each other and perpendicular to the longitudinal center plane of the vehicle;
• During steering, all 4 wheels must move in a circular motion around an instantaneous center point.

When the Ackerman model turns, the inner and outer wheels turn at different angles, and the turning radius of the inner measured tire is less than that of the outer tire, as shown in the following diagram.

Where β is the front outer wheel angle of the trolley, α is the front inner wheel angle of the trolley, K is the wheelbase, L is the wheelbase, and R is the minimum turning radius, which is a very important parameter in Ackermann's motion model. This leads to the Ackermann core equation:

Minimum turning radius R:

Low-speed environments.

The turning radius of the vehicle's travel path changes slowly, at which point we can assume that the rate of change of the vehicle's direction is equal to the vehicle's angular velocity. Then the angular velocity of the vehicle is

Using the relationship between the wheelbase L and the front wheel angle of rotation θ yields that

When the slip angle (the angle between the direction in which the car is traveling and the direction in which the wheel rims are pointing) is small and the rear wheel deflection is zero, combining 1.1 and 1.2 yields that

As mentioned above the inside and outside tires have different steering radii.

Then it can be found that the front wheel angle of rotation, θ, is that

Therefore, the minimum turning radius R is that

The position of the vehicle is predicted based on the minimum turning radius:

Assuming that the vehicle speed is v, the distance traveled in time t is v*t, and the direct turning radius is R in the vehicle coordinate system, it can be deduced that the vehicle has turned at an angle σ, the

Combined with the turning radius R calculated above, it follows that the

Then, in the vehicle coordinate system, the coordinates of the vehicle position after moment t are updated as.