transformtraj

Generate trajectories between two transformations

Syntax

[tforms,vel,acc] = transformtraj(T0,TF,tInterval,tSamples)

[tforms,vel,acc] = transformtraj(T0,TF,tInterval,tSamples,Name=Value)

Description

[tforms,vel,acc] = transformtraj(T0,TF,tInterval,tSamples) generates a trajectory that interpolates between two homogeneous transformations, T0 and TF, with points based on the time interval and given time samples.

[tforms,vel,acc] = transformtraj(T0,TF,tInterval,tSamples,Name=Value) specifies additional parameters using name-value arguments.

Examples

Interpolate Between Homogenous Transformations

Build transformations from two orientations and positions. Specify the time interval and vector of times for interpolating.

t0 = axang2tform([0 1 1 pi/4])*trvec2tform([0 0 0]);
tF = axang2tform([1 0 1 6*pi/5])*trvec2tform([1 1 1]);
tInterval = [0 1];
tvec = 0:0.01:1;

Interpolate between the points. Plot the trajectory using plotTransforms. Convert the transformations to quaternion rotations and linear transitions. The figure shows all the intermediate transformations of the coordinate frame.

[tfInterp, v1, a1] = transformtraj(t0,tF,tInterval,tvec);

rotations = tform2quat(tfInterp);
translations = tform2trvec(tfInterp);

plotTransforms(translations,rotations)
xlabel('X')
ylabel('Y')
zlabel('Z')

Interpolate Between SE(3) Transformations

Build the first transformation from an axis-angle rotation and build the final transformation from both an axis-angle rotation and a translation.

t0 = se3([0 1 1 pi/4],"axang");
tF = se3([1 0 1 6*pi/5],"axang")*se3([1 1 1],"trvec"); 

Specify the time interval and vector of times for interpolating.

tInterval = [0 1];
tvec = 0:0.01:1; 

Interpolate between the transformations. Plot the trajectory using plotTransforms.

[tfInterp] = transformtraj(t0,tF,tInterval,tvec); 
plotTransforms(tfInterp(1:10:end))

Input Arguments

T0 — Initial transformation

4-by-4 homogeneous transformation | se3 object

Initial transformation, specified as a 4-by-4 homogeneous transformation or a scalar se3 object. The function generates a trajectory that starts at the initial transformation, T0, and goes to the final transformation, TF.

T0 and TF must be of the same type. For example, if T0 is a scalar se3 object, then TF must be a scalar se3 object.

Data Types: single | double

TF — Final transformation

4-by-4 homogeneous transformation | se3 object

Final transformation, specified as a 4-by-4 homogeneous transformation or a scalar se3 object. The function generates a trajectory that starts at the initial transformation, T0, and goes to the final transformation, TF.

T0 and TF must be of the same type. For example, if T0 is a scalar se3 object, then TF must be a scalar se3 object.

Data Types: single | double

tInterval — Start and end times for trajectory

two-element vector

Start and end times for the trajectory, specified as a two-element vector in seconds.

Example: [0 10]

Data Types: single | double

tSamples — Time samples for trajectory

m-element vector

Time samples for the trajectory, specified as an m-element vector in seconds.

Example: 0:0.01:10

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: TimeScaling=[0 1 1; 0 1 0; 0 0 0]

TimeScaling — Time scaling vector and first two derivatives

3-by-m vector

Time scaling vector and the first two derivatives, specified as a 3-by-m vector, where m is the length of tSamples. By default, the time scaling is a linear time scaling between the time points in tInterval.

For a nonlinear time scaling, specify the values of the time points as position time scaling in the first row. The second and third rows are the first and second derivative of the first row, 1/s and 1/s², respectively. For example, to follow the path with a linear velocity to the halfway point, and then jump to the end, the time-scaling would be:

s(1,:) = [0 0.25 0.5 1 1 1] % Position time scaling, s(t)
s(2,:) = [1    1   1 0 0 0] % Velocity time scaling, ds/dt
s(3,:) = [0    0   0 0 0 0] % Acceleration time scaling, d^2s/dt^2

All elements in the first row must be in the range [0, 1].

Data Types: single | double

Output Arguments

tforms — Transformation trajectory

4-by-4-by-m homogeneous transformation matrix array | m-element array of se3 objects

Transformation trajectory, returned as a 4-by-4-by-m homogeneous transformation matrix array or an m-element array of se3 objects. m is the number of points in tSamples.

vel — Transformation velocities

6-by-m matrix

Transformation velocities, returned as a 6-by-m matrix in m/s, where m is the number of points in tSamples. The first three elements are the angular velocities, and the second three elements are the velocities in time.

acc — Transformation accelerations

6-by-m matrix

Transformation accelerations, returned as a 6-by-m matrix in m/s², where m is the number of points in tSamples. The first three elements are the angular accelerations, and the second three elements are the accelerations in time.

References

[1] Lynch, Kevin M., and Frank C. Park. Modern Robotics: Mechanics, Planning, and Control. Cambridge University Press, 2017.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2019a

R2023a: transformtraj supports SE(3) transformation object inputs

The T0 and TF arguments of transformtraj now accept se3 objects and the resulting output value for the tforms argument is an m-element array of se3 objects. m is the number of points in tSamples.