Wheeled Mobile Robot Kinematics

This page introduces the kinematics of different wheeled mobile robots. For further reference see Siciliano et.al - Robotics: Modelling, Planning and Control and Kevin M. Lynch and Frank C. Park - Modern Robotics: Mechanics, Planning, And Control.

Wheeled mobile robots can be classified in two categories:

Omnidirectional robots

which can move instantaneously in any direction in the plane, and

Nonholonomic robots

which cannot move instantaneously in any direction in the plane.

The forward integration of the kinematic model using the encoders of the wheel actuators — is referred to as odometric localization or passive localization or dead reckoning. We will call it just odometry.

Omnidirectional Wheeled Mobile Robots

Omnidirectional Drive Robots using Omni Wheels

The below explains the kinematics of omnidirectional drive robots using 3 or more omni wheels.It follows the coordinate conventions defined in ROS REP 103.

• 
  

  
  

• is the number of wheels.

Inverse Kinematics

The necessary angular velocity of the wheels to achieve a desired body twist can be calculated using the below matrix:

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.

• .

Each swerve module

relative to the center, typically:

• Front-left:

Front-right:

Back-left:

Back-right:

Inverse Kinematics

For each module

, the velocity vector is:

are:

Odometry

The body twist of the robot is computed from the wheel velocities

. Each module’s velocity components in the body frame are:

The chassis velocities are calculated as:

) 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

• 
  

• -axis.

In the following, we want to command the robot with a desired body twist

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

• axis.

as desired system inputs. The forward kinematics of the unicycle can be calculated with

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.

• is the wheel track (the distance between the wheels).

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:

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 wheelbase.

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.

• is the wheel track of the rear axle.

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.

• is the wheel track of the front axle, measured between the two kingpins.

To prevent the front wheels from slipping, the steering angle of the front wheels cannot be equal.This is the so-called Ackermann steering.

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.

• is the distance from the kingpin to the contact point of the front wheel with the ground.

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.,