# 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 using plus in Java, or the + operator in C++. Addition adds the two vectors.

• Subtraction: Subtraction between two Translation2d can be performed using minus in Java, or the binary - operator in C++. Subtraction subtracts the two vectors.

• Multiplication: Multiplication of a Translation2d and a scalar can be performed using times in Java, or the * operator in C++. This multiplies the vector by the scalar.

• Division: Division of a Translation2d and a scalar can be performed using div in Java, or the / operator in C++. 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 using rotateBy. 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 using unaryMinus in Java, or the unary - operator in C++.

## Rotation2d

Transformations for Rotation2d are just arithmetic operations on the angle measure represented by the Rotation2d.

• plus (Java) or + (C++): Adds the rotation component of other to this Rotation2d’s rotation component

• minus (Java) or binary - (C++): Subtracts the rotation component of other to this Rotation2d’s rotation component

• unaryMinus (Java) or unary - (C++): Multiplies the rotation component by a scalar of -1.

• times (Java) or * (C++) : Multiplies the rotation component by a scalar.

## Transform2d and Twist2d

WPILib provides 2 classes, Transform2d (Java, C++), which represents a transformation to a pose, and Twist2d (Java, C++) 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.

Nota

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.