Transformations
Translation2d
Operations on a Translation2d
perform operations to the vector represented by the Translation2d
.
Addition: Addition between two
Translation2d
a and b can be performed usingplus
in Java, or the+
operator in C++/Python. Addition adds the two vectors.Subtraction: Subtraction between two
Translation2d
can be performed usingminus
in Java, or the binary-
operator in C++/Python. Subtraction subtracts the two vectors.Multiplication: Multiplication of a
Translation2d
and a scalar can be performed usingtimes
in Java, or the*
operator in C++/Python. This multiplies the vector by the scalar.Division: Division of a
Translation2d
and a scalar can be performed usingdiv
in Java, or the/
operator in C++/Python. This divides the vector by the scalar.Rotation: Rotation of a
Translation2d
by a counter-clockwise rotation \(\theta\) about the origin can be performed by usingrotateBy
. This is equivalent to multiplying the vector by the matrix \(\begin{bmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{bmatrix}\)Additionally, you can rotate a
Translation2d
by 180 degrees by usingunaryMinus
in Java, or the unary-
operator in C++/Python.
Rotation2d
Transformations for Rotation2d
are just arithmetic operations on the angle measure represented by the Rotation2d
.
plus
(Java) or+
(C++/Python): Adds the rotation component ofother
to thisRotation2d
’s rotation componentminus
(Java) or binary-
(C++/Python): Subtracts the rotation component ofother
to thisRotation2d
’s rotation componentunaryMinus
(Java) or unary-
(C++/Python): Multiplies the rotation component by a scalar of -1.times
(Java) or*
(C++/Python) : Multiplies the rotation component by a scalar.
Transform2d and Twist2d
WPILib provides 2 classes, Transform2d
(Java, C++, Python
>`, which represents a transformation to a pose, and Twist2d
([Java] (https://github.wpilib.org/allwpilib/docs/development/java/edu/wpi/first/math/geometry/Twist2d.html), C++, Python
) which represents a movement along an arc. Transform2d
and Twist2d
all have x, y and \(\theta\) components.
Transform2d
represents a relative transformation. It has an translation and a rotation component. Transforming a Pose2d
by a Transform2d
rotates the translation component of the transform by the rotation of the pose, and then adds the rotated translation component and the rotation component to the pose. In other words, Pose2d.plus(Transform2d)
returns \(\begin{bmatrix} x_p \\ y_p \\ \theta_p \end{bmatrix}+\begin{bmatrix} cos\theta_p & -sin\theta_p & 0 \\ sin\theta_p & cos\theta_p & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}x_t \\ y_t \\ \theta_t \end{bmatrix}\)
Twist2d
represents a change in distance along an arc. Usually, this class is used to represent the movement of a drivetrain, where the x component is the forward distance driven, the y component is the distance driven to the side (left positive), and the \(\theta\) component is the change in heading. The underlying math behind finding the pose exponential (new pose after moving the pose forward along the curvature of the twist) can be found here in chapter 10.
Note
For nonholonomic drivetrains, the y component of a Twist2d
should always be 0.
Both classes can be used to estimate robot location. Twist2d is used in WPILib’s odometry classes to update the robot’s pose based on movement, while Transform2d can be used to estimate the robot’s global position from vision data.