Rotation Trajectory

Generate trajectory between two orientations

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Libraries:
Robotics System Toolbox / Utilities

Description

The Rotation Trajectory block generates an interpolated trajectory between two rotation matrices. The block outputs the rotation at the times given by the Time input, which can be a scalar or vector.

The trajectory is computed using quaternion spherical linear interpolation (SLERP) and finds the shortest path between points. Select the Use custom time scaling check box to compute using a custom time scaling. The block uses linear time scaling by default.

The initial and final values are held constant outside the time period defined in the Time interval parameter.

Examples

Generate Rotation Trajectory

Generate a trajectory that interpolates between rotations using the Rotation Trajectory block in

• Open Live Script

Use Custom Time Scaling for a Rotation Trajectory

Specify custom time-scaling in the Rotation Trajectory block in Simulink^® to execute an interpolated trajectory.

• Open Live Script

Ports

Input

Time — Time point along trajectory

scalar | vector

Time point along the trajectory, specified as a scalar or vector. In general, when specified as a scalar, this value is synced with simulation time and is used to specify the time point for sampling the trajectory. The block outputs a vector of the trajectory variables at that instant in time. If the time is specified as a vector, the block outputs a matrix with each column corresponding to each element of the vector.

Data Types: single | double

R0 — Initial orientation

four-element quaternion vector | 3-by-3 rotation matrix

Initial orientation, specified as a four-element quaternion vector or 3-by-3 rotation matrix. The function generates a trajectory that starts at the initial orientation, R0, and goes to the final orientation, RF.

Example: [1 0 0 0]'

Dependencies

To enable this input, set the Waypoint source to External.

To specify quaternions, set Rotation Format parameter to Quaternion.

To specify rotation matrices, set Rotation Format parameter to Rotation.

Data Types: single | double

RF — Final orientation

four-element vector | 3-by-3 rotation matrix

Initial orientation, specified as a four-element vector or 3-by-3 rotation matrix. The function generates a trajectory that starts at the initial orientation, R0, and goes to the final orientation, RF.

Example: [0 0 1 0]'

Dependencies

To enable this input, set the Waypoint source to External.

To specify quaternions, set Rotation Format parameter to Quaternion.

To specify rotation matrices, set Rotation Format parameter to Rotation.

Data Types: single | double

TimeInterval — Start and end times for trajectory

two-element vector

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

Example: [0 10]

Dependencies

To enable this input, set the Waypoint source to External.

Data Types: single | double

TSTime — Time scaling time points

scalar | p-element vector

Time scaling time points, specified as a scalar or n p-element vector, where p is the number of points for time scaling. By default, the time scaling is a linear time scaling spanning the TimeInterval. Specify the actual time scaling values in TimeScaling.

If the Time input is specified at a time not specified by these points, interpolation is used to find the right scaling time.

Dependencies

To enable this parameter, select the Use custom time scaling check box and set Parameter source to External.

To specify a scalar, the Time input must be a scalar.

Data Types: single | double

TimeScaling — Time scaling vector and first two derivatives

three-element vector | 3-by-p matrix

Time scaling vector and its first two derivatives, specified as a three element vector or a 3-by-p matrix, where m is the length of TSTime. By default, the time scaling is a linear time scaling spanning the TimeInterval.

For a nonlinear time scaling, specify the values of the time points in the first row. The second and third rows are the velocity and acceleration of the time points, 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
s(2,:) = [1    1   1 0 0 0] % Velocity
s(3,:) = [0    0   0 0 0 0] % Acceleration

Dependencies

To enable this parameter, select the Use custom time scaling check box and set Parameter source to External.

To specify a three-element vector, the Time and TSTime inputs must be a scalar.

Data Types: single | double

Output

R — Orientation vectors

4-by-m quaternion array | 3-by-3-by-m rotation matrix array

Orientation vectors, returned as a 4-by-m quaternion array or 3-by-3-by-m rotation matrix array, where m is the number of points in the input to Time.

Dependencies

To get a quaternion array, set Rotation Format parameter to Quaternion.

To get a rotation matrix array, set Rotation Format parameter to Rotation.

omega — Orientation angular velocity

3-by-m matrix

Orientation angular velocity, returned as a 3-by-m matrix, where m is the number of points in the input to Time.

alpha — Orientation angular acceleration

3-by-m matrix

Orientation angular acceleration, returned as a 3-by-m matrix, where m is the number of points in the input to Time.

Parameters

Rotation format — Format for orientations

Quaternion (default) | Rotation Matrix

Select Rotation Matrix to specify the Initial rotation and Final rotation as 3-by-3 rotation matrices and get the orientation output (port R) as a rotation matrix array. By default, the initial and final rotations are specified as four-element quaternion vectors.

Waypoint source — Source for waypoints

Internal (default) | External

Specify External to specify the Initial rotation, Final rotation, and Time interval parameters as block inputs instead of block parameters.

Initial rotation — Initial orientation

[1 0 0 0]' (default) | four-element quaternion vector | 3-by-3 rotation matrix

Initial orientation, specified as a four-element quaternion vector or 3-by-3 rotation matrix. The function generates a trajectory that starts at the Initial rotation and goes to the Final rotation.

Dependencies

To specify quaternions, set Rotation Format parameter to Quaternion.

To specify rotation matrices, set Rotation Format parameter to Rotation.

Data Types: single | double

Final rotation — Final orientation

[0 0 1 0]' (default) | four-element vector | 3-by-3 rotation matrix

Final orientation, specified as a four-element vector or 3-by-3 rotation matrix. The function generates a trajectory that starts at the Initial rotation and goes to the Final rotation.

Dependencies

To specify quaternions, set Rotation Format parameter to Quaternion.

To specify rotation matrices, set Rotation Format parameter to Rotation.

Data Types: single | double

Time interval — Start and end times for trajectory

[0 10] (default) | two-element vector

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

Data Types: single | double

Use custom time scaling — Enable custom time scaling

off (default) | on

Enable to specify custom time scaling for the trajectory using the Parameter Source, Time scaling time, and Time scaling values parameters.

Parameter source — Source for waypoints

Internal (default) | External

Specify External to specify the Time scaling time and Time scaling values parameters as block inputs instead of block parameters.

Dependencies

To enable this parameter, select the Use custom time scaling check box.

Time scaling time — Time scaling time points

2:0.1:3 (default) | scalar | p-element vector

Time scaling time points, specified as a scalar or p-element vector, where p is the number of points for time scaling. By default, the time scaling is a linear time scaling spanning the Time interval. Specify the actual time scaling values in Time scaling values.

If the Time input is specified at a time not specified by these points, interpolation is used to find the right scaling time.

Dependencies

To enable this parameter, select the Use custom time scaling check box.

To specify a scalar, the Time input must be a scalar.

Data Types: single | double

Time scaling values — Time scaling vector and first two derivatives

[0:0.1:1; ones(1,11); zeros(1,11)] (default) | three-element vector | 3-by-m matrix

Time scaling vector and its first two derivatives, specified as a three-element vector or 3-by-p matrix, where p is the length of Time scaling time. By default, the time scaling is a linear time scaling spanning the Time interval.

For a nonlinear time scaling, specify the values of the time points in the first row. The second and third rows are the velocity and acceleration of the time points, 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
s(2,:) = [1    1   1 0 0 0] % Velocity
s(3,:) = [0    0   0 0 0 0] % Acceleration

Dependencies

To enable this parameter, select the Use custom time scaling check box.

To specify a three-element vector, the Time and TSTime inputs must be a scalar.

Data Types: single | double

Simulate using — Type of simulation to run

Interpreted execution (default) | Code generation

• Interpreted execution — Simulate model using the MATLAB^® interpreter. This option shortens startup time but has a slower simulation speed than Code generation. In this mode, you can debug the source code of the block.

• Code generation — Simulate model using generated C code. The first time you run a simulation, Simulink generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time, but the speed of the subsequent simulations is comparable to Interpreted execution.

Tunable: No

Tips

For better performance, consider these options:

• Minimize the number of waypoint or parameter changes.

• Set the Waypoint source parameter to Internal.

• Set the Simulate using parameter to Code generation. For more information, see Interpreted Execution vs. Code Generation (Simulink).

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 Simulink® Coder™.

Version History

Introduced in R2019a