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Frames & orientations

Convert W, P, R angles to quaternions, interpolate orientations, and change frames between flange, tool, user frame and world frame.

Quaternions
Change of frame
API reference

FANUC controllers give orientations as W, P, R angles, in degrees: rotation W around X, then P around Y, then R around Z, all around the axes of the reference frame. XYZWPRPosition gives conversions to quaternions and the usual frame operations. These functions work offline and can be used with any protocol of the SDK.

W, P and R: rotations around the X, Y and Z axes of the reference frame.

Quaternions

Quaternions are easier than angles to interpolate and to compare orientations.

static void Main()
{
var position = new XYZWPRPosition(500, 0, 300, 180, 0, 45);

// W, P, R angles to quaternion, and back
Quaternion q = position.GetQuaternion();
position.SetQuaternion(Quaternion.FromAxisAngle(0, 0, 1, 90));   // 90 degrees around Z

// Interpolation and angle between two orientations
Quaternion a = new XYZWPRPosition(0, 0, 0, 180, 0, 0).GetQuaternion();
Quaternion b = new XYZWPRPosition(0, 0, 0, 180, 30, 0).GetQuaternion();
Quaternion halfWay = Quaternion.Slerp(a, b, 0.5);
double angle = a.AngleTo(b);                // 30 degrees
double[] axisAngle = halfWay.ToAxisAngle(); // x, y, z, angle in degrees
}
}

Click to see the full code

GetQuaternion() and SetQuaternion() convert between W, P, R and a quaternion (Qw, Qx, Qy, Qz).
Quaternion.Slerp() interpolates on the shortest way between two orientations.
AngleTo() gives the angle between two orientations, in degrees.
FromAxisAngle(), ToAxisAngle(), FromRotationMatrix() and ToRotationMatrix() convert to and from other representations.

Change of frame

A position is a frame: its X, Y, Z give the origin and its W, P, R the orientation. The same functions work for the positions of the robot, tool frames and user frames.

static void Main()
{
var tool = new XYZWPRPosition(0, 0, 150, 0, 0, 0);            // tool frame, relative to the flange
var userFrame = new XYZWPRPosition(800, -200, 0, 0, 0, 90);    // user frame, relative to the world frame
var flange = new XYZWPRPosition(700, 0, 400, 180, 0, 0);       // flange in the world frame

// Flange <-> tool center point
XYZWPRPosition tcp = flange.FlangeToTcp(tool);
XYZWPRPosition flangeAgain = tcp.TcpToFlange(tool);

// World frame <-> user frame
XYZWPRPosition tcpInUserFrame = tcp.WorldToUserFrame(userFrame);
XYZWPRPosition tcpInWorld = tcpInUserFrame.UserFrameToWorld(userFrame);

// General frame operations
XYZWPRPosition composed = userFrame.Multiply(tcpInUserFrame);   // same as tcpInWorld
XYZWPRPosition inverse = userFrame.Inverse();
double[,] matrix = flange.ToHomogeneousMatrix();                // 4 x 4
}
}

Click to see the full code

World frame, user frame, flange and tool center point (TCP).

Need	Method
Position of the tool center point from the flange position	`flange.FlangeToTcp(tool)`
Flange position from the tool center point	`tcp.TcpToFlange(tool)`
Position in the world frame from a position in a user frame	`position.UserFrameToWorld(userFrame)`
Position in a user frame from a position in the world frame	`position.WorldToUserFrame(userFrame)`
Composition of two frames (A x B)	`a.Multiply(b)`
Inverse frame	`frame.Inverse()`
4 x 4 homogeneous matrix	`position.ToHomogeneousMatrix()`

The motion planner does these conversions for you when ToolFrame and UserFrame are set (see Joint & Cartesian motions).

API reference

Class

Quaternion

C#Python

Quaternion that represents an orientation (Qw + Qx.i + Qy.j + Qz.k). Use GetQuaternion and SetQuaternion(Quaternion) to convert from and to W, P, R angles.

Member	Type	Description
Quaternion() Constructor		Creates the identity quaternion (no rotation)
Quaternion(double, double, double, double) Constructor		Creates a quaternion from its components qw : Scalar part qx : X component of the vector part qy : Y component of the vector part qz : Z component of the vector part
Norm Property read only	double	Norm of the quaternion (1 for a rotation)
Qw Property	double	Scalar part
Qx Property	double	X component of the vector part
Qy Property	double	Y component of the vector part
Qz Property	double	Z component of the vector part
AngleTo(Quaternion) Method	double	Angle of the rotation between the two orientations, in degrees (0 to 180) other : Other orientation
Conjugate() Method	Quaternion	Returns the conjugate of this quaternion. For a rotation, it is the inverse rotation.
Dot(Quaternion) Method	double	Dot product of the two quaternions other : Other quaternion
Equals(object) Method	bool
FromAxisAngle(double, double, double, double) Method static	Quaternion	Creates a rotation around an axis x : X component of the axis y : Y component of the axis z : Z component of the axis angle : Rotation angle in degrees
FromRotationMatrix(double[,]) Method static	Quaternion	Creates a quaternion from a rotation matrix (3x3, or 4x4 homogeneous matrix) matrix : Rotation matrix
GetHashCode() Method	int
Multiply(Quaternion) Method	Quaternion	Returns the product this x other: the rotation other applied after the rotation this, in the frame of this. other : Right operand
Normalize() Method	Quaternion	Returns this quaternion with a norm of 1
Slerp(Quaternion, Quaternion, double) Method static	Quaternion	Spherical linear interpolation between two orientations, on the shortest way start : Orientation for t = 0 end : Orientation for t = 1 t : Interpolation parameter, usually between 0 and 1
ToAxisAngle() Method	double[]	Returns the rotation axis and angle: [x, y, z, angle in degrees]. The axis is a unit vector and the angle is between 0 and 180.
ToRotationMatrix() Method	double[,]	Returns the 3x3 rotation matrix of this orientation
ToString() Method	string

Class

XYZWPRPositioninherits XYZPosition

C#Python

Cartesian position X, Y, Z with W, P, R rotations

Member	Type	Description
XYZWPRPosition() Constructor		Default constructor
XYZWPRPosition(double, double, double, double, double, double) Constructor		Constructor with position and rotations
P Property	double	P rotation in degrees (Ry)
R Property	double	R rotation in degrees (Rz)
W Property	double	W rotation in degrees (Rx)
Equals(object) Method	bool
FlangeToTcp(XYZWPRPosition) Method	XYZWPRPosition	Converts a flange position to the position of the tool center point (TCP) tool : Tool frame, relative to the flange (for example a UTOOL value)
GetHashCode() Method	int
GetQuaternion() Method	Quaternion	Returns the orientation W, P, R as a quaternion
Inverse() Method	XYZWPRPosition	Returns the inverse of this frame
Multiply(XYZWPRPosition) Method	XYZWPRPosition	Returns the composition of this frame with another one: the pose `other`, expressed in this frame, converted to the frame where this position is expressed. other : Pose expressed in this frame
SetQuaternion(Quaternion) Method	void	Sets the orientation W, P, R from a quaternion. Angles are between -180 and 180 degrees. quaternion : Orientation
TcpToFlange(XYZWPRPosition) Method	XYZWPRPosition	Converts a position of the tool center point (TCP) to the flange position tool : Tool frame, relative to the flange (for example a UTOOL value)
ToHomogeneousMatrix() Method	double[,]	Convert position to a homogeneous rotation and translation 4x4 matrix
ToString() Method	string
UserFrameToWorld(XYZWPRPosition) Method	XYZWPRPosition	Converts this position, expressed in a user frame, to the world frame userFrame : User frame, relative to the world frame (for example a UFRAME value)
WorldToUserFrame(XYZWPRPosition) Method	XYZWPRPosition	Converts this position, expressed in the world frame, to a user frame userFrame : User frame, relative to the world frame (for example a UFRAME value)

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Frames & orientations

Quaternions

Change of frame

API reference