https://control.ros.org/rolling/doc/ros2_controllers/doc/mobile_robot_kinematics.html
is the robot’s body-frame coordinate system, located at the contact point of the wheel on the ground.
is the world coordinate system.
is the robot’s linear velocity on the x-axis.
is the robot’s linear velocity on the y-axis.
is the robot’s angular velocity on the z-axis.
is the robot’s radius / the distance between the robot’s center and the wheels.
Red arrows on the wheel
signify the positive direction of their rotation
is the angular offset of the first wheel from .
is the angle between each wheel which can be calculated using the below equation where
Inverse Kinematics
The necessary angular velocity of the wheels to achieve a desired body twist can be calculated using the below matrix:
Here are the angular velocities of the wheels and is the radius of the wheels.These equations can be written in algebraic form for any wheel like this:
Forward Kinematics
The body twist of the robot can be obtained from the wheel velocities by using the pseudoinverse of matrix
.
Swerve Drive Robots
The below explains the kinematics of omnidirectional drive robots using four swerve modules, each with independently controlled steering and driving motors. It follows the coordinate conventions defined in REP-103.
is the robot’s body-frame coordinate system, located at the geometric center of the robot.
is the world coordinate system.
is the robot’s linear velocity on the x-axis.
is the robot’s linear velocity on the y-axis.
is the robot’s angular velocity on the z-axis.
is the wheelbase (distance between front and rear wheels).
is the track width (distance between left and right wheels).
Red arrows on wheel
signify the direction of the wheel’s velocity
Each swerve module
, for (typically front-left, front-right, back-left, back-right) is located at relative to the center, typically:
Front-right:
Back-left:
Back-right:
Inverse Kinematics
For each module
at position , the velocity vector is:
The wheel velocity and steering angle are:
Odometry
The body twist of the robot is computed from the wheel velocities
and steering angles . Each module’s velocity components in the body frame are:
The chassis velocities are calculated as:
Odometry updates the robot’s pose (, , ) in the global frame using the computed chassis velocities. The global velocities are:
Nonholonomic Wheeled Mobile Robots
Unicycle model
To define the coordinate systems (ROS coordinate frame conventions, the coordinate systems follow the right-hand rule), consider the following simple unicycle model
is the robot’s body-frame coordinate system, located at the contact point of the wheel on the ground.
is the world coordinate system.
are the robot’s Cartesian coordinates in the world coordinate system.
is the robot’s heading angle, i.e. the orientation of the robot’s -axis w.r.t. the world’s
In the following, we want to command the robot with a desired body twist
where is the linear velocity of the robot in its body-frame, and is the angular velocity of the robot in its body-frame. As we consider steering robots on a flat surface, it is sufficient to give
, i.e. the linear velocity of the robot in direction of the axis.
, i.e. the angular velocity of the robot about the
as desired system inputs. The forward kinematics of the unicycle can be calculated with
We will formulate the inverse kinematics to calculate the desired commands for the robot (wheel speed or steering) from the given body twist.
Differential Drive Robot
Citing Siciliano et.al - Robotics: Modelling, Planning and Control:
A unicycle in the strict sense (i.e., a vehicle equipped with a single wheel)is a robot with a serious problem of balance in static conditions. However,there exist vehicles that are kinematically equivalent to a unicycle but morestable from a mechanical viewpoint.
One of these vehicles is the differential drive robot, which has two wheels, each of which is driven independently.
Forward Kinematics
The forward kinematics of the differential drive model can be calculated from the unicycle model above using
Inverse Kinematics
The necessary wheel speeds to achieve a desired body twist can be calculated with:
Odometry
We can use the forward kinematics equations above to calculate the robot’s odometry directly from the encoder readings.
Car-Like (Bicycle) Model
The following picture shows a car-like robot with two wheels, where the front wheel is steerable. This model is also known as the bicycle model.
is the steering angle of the front wheel, counted positive in direction of rotation around -axis.
is the velocity of the rear and front wheel.
We assume that the wheels are rolling without slipping. This means that the velocity of the contact point of the wheel with the ground is zero and the wheel’s velocity points in the direction perpendicular to the wheel’s axis. The Instantaneous Center of Rotation (ICR), i.e. the center of the circle around which the robot rotates, is located at the intersection of the lines that are perpendicular to the wheels’ axes and pass through the contact points of the wheels with the ground.
As a consequence of the no-slip condition, the velocity of the two wheels must satisfy the following constraint:
Forward Kinematics
The forward kinematics of the car-like model can be calculated with
Inverse Kinematics
The steering angle is one command input of the robot:
For the rear-wheel drive, the velocity of the rear wheel is the second input of the robot:
For the front-wheel drive, the velocity of the front wheel is the second input of the robot:
Odometry
We have to distinguish between two cases: Encoders on the rear wheel or on the front wheel.
For the rear wheel case:
For the front wheel case:
Double-Traction Axle
The following image shows a car-like robot with three wheels, with two independent traction wheels at the rear.
Forward Kinematics
The forward kinematics is the same as the car-like model above.
Inverse Kinematics
The turning radius of the robot is
Then the velocity of the rear wheels must satisfy these conditions to avoid skidding
Odometry
The calculation of
from two encoder measurements of the traction axle is overdetermined.If there is no slip and the encoders are ideal,
holds. But to get a more robust solution, we take the average of both , i.e.,
Ackermann Steering
The following image shows a four-wheeled robot with two independent steering wheels in the front.
To prevent the front wheels from slipping, the steering angle of the front wheels cannot be equal.This is the so-called Ackermann steering.
Note
Ackermann steering can also be achieved by a mechanical linkage between the two front wheels. In this case the robot has only one steering input, and the steering angle of the two front wheels is mechanically coupled. The inverse kinematics of the robot will then be the same as in the car-like model above.
Forward Kinematics
The forward kinematics is the same as for the car-like model above.
Inverse Kinematics
The turning radius of the robot is
Then the steering angles of the front wheels must satisfy these conditions to avoid skidding
Odometry
The calculation of
from two angle measurements of the steering axle is overdetermined.If there is no slip and the measurements are ideal,
holds. But to get a more robust solution, we take the average of both , i.e.,
Ackermann Steering with Traction
The following image shows a four-wheeled car-like robot with two independent steering wheels at the front, which are also driven independently.
Forward Kinematics
The forward kinematics is the same as the car-like model above.
Inverse Kinematics
To avoid slipping of the front wheels, the velocity of the front wheels cannot be equal and
with turning radius of the robot and the left/right front wheel
This results in the following inverse kinematics equations
with the steering angles of the front wheels from the Ackermann steering equations above.
Odometry
The calculation of
from two encoder measurements of the traction axle is again overdetermined.If there is no slip and the encoders are ideal,
holds. But to get a more robust solution, we take the average of both , i.e.,