Robot Kinematics
Kinematics is the study of the geometry of motion. Kinematics is the branch of classical mechanics that describes various aspects of motion such as velocity, acceleration, displacement, time, and trajectory. There is no description of the forces that cause the motion to happen. The word “kinematics” comes from a Greek word “kinesis” meaning motion, and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion).
In robotics we are mostly interested in the motion of groups of objects that are connected, or jointed, together into a system. When objects are connected together they can be subject to certain constraints that restrict their movement. For wheeled robots we are interested in determining how the robot can move with reference to the constraints imposed by the motion of the wheels. So that's where we'll start.
We want to examine the constraints that exist by the connection of the wheels to the robot. A wheel is connected to an axel which is typically driven by a motor. The motor causes the wheel to rotate which produces forward, or backward, motion of the robot. Nothing too suprising there! What is of note though, is that the wheel cannot move sideways to its direction of motion, so here's where we find our first constraint.
Our next constraint comes in when we have more than one wheel on the robot, which is of course the common configuration. When we have multiple wheels we must ensure that they have a common Instantaineous Center of Curvature (ICC). This means that the wheels must be connected in a way that they are all aligned towards the same point. If this is not the case then the wheels will slip in a sideways direction making it extremely difficult to control the motion of the robot. This is illustrated below. Typically, there are two wheels on the same axel which causes them to be aligned towards the same center point. However, with a 4-wheeled robot there are two sets of wheels and two axels which means that the back wheels will always slip. These are called skid drive robots.
Differential Drive Robots
The most common robot configuration is the Differential Drive robot, so let's look at its kinematics. This robot will have two wheels each connected to a motor. The wheels will be aligned on a common axis. If both motors rotate forward or backward at the same speed then the robot will move in a straight line. If one motor rotates faster than the other then the robot will take a curved path. One very useful feature of this robot is its ability to rotate on the spot which enables it to move in tight spaces. This is acheived by rotating each motor in a different direction.
Let's look at some of the mathematics behind differential drive robot kinematics. While we can vary the velocity of each wheel, for the robot to perform rolling motion, the robot must rotate about a point that lies along its common left and right wheel axis. The point that the robot rotates about is known as the ICC - Instantaneous Center of Curvature.
By varying the velocities of the two wheels, we can vary the trajectories that the robot takes. Because the rate of rotation ω about the ICC must be the same for both wheels, we can write the equations shown at the top right of the diagram, where
At any instance in time we can compute the distance to the ICC
These formulars are used in the DifferentialDriveKinematics class of the WPI library.
Constraints
Constraints can be position constraints (something in the way), control constraints (limited amount of torque), model constraints (system kinematics, holonomic constraint).
A nonholonomic constraint does not restrain the possible configurations of the system, but rather the manner in which those configurations can be reached. While a holonomic constraint reduces the number of degrees of freedom of a system by one, a nonholonomic constraint does not.
Swerve Drive Robots
The function toSwerveModuleStates() performs inverse kinematics to get the module states from a desired chassis velocity.
The function toChassisSpeeds() performs forward kinematics to return the resulting chassis state from the given module states. This method is often used for odometry -- determining the robot's position on the field using data from the real-world speed and angle of each module on the robot.
The function desaturateWheelSpeeds() renormalizes the wheel speeds if any individual speed is above the specified maximum. Sometimes, after inverse kinematics, the requested speed from one or more modules may be above the max attainable speed for the driving motor on that module. To fix this issue, one can reduce all the wheel speeds to make sure that all requested module speeds are at-or-below the absolute threshold, while maintaining the ratio of speeds between modules.
References
- FRC Documentation - Kinematics and Odometry