Polynomial Trajectory
Generate polynomial trajectories through waypoints
Libraries:
Robotics System Toolbox /
Utilities
Description
The Polynomial Trajectory block generates trajectories to travel through waypoints at the given time points using either cubic, quintic, or B-spline polynomials. The block outputs positions, velocities, and accelerations for achieving this trajectory based on the Time input. For B-spline polynomials, the waypoints actually define the control points for the convex hull of the B-spline instead of the actual waypoints, but the first and last waypoint are still met.
The initial and final values are held constant outside the time period defined in Time points.
Examples
Generate Cubic Polynomial Trajectory
Generate a cubic polynomial trajectory using the Polynomial Trajectory block in Simulink®.
Generate B-Spline Trajectory
Generate a B-spline trajectory using the Polynomial Trajectory block in Simulink.
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
Waypoints — Waypoint positions along trajectory
n-by-p matrix
Positions of waypoints of the trajectory at given time points, specified as an
n-by-p matrix, where n is
the dimension of the trajectory and p is the number of waypoints.
If you specify the Method as B-spline
, these
waypoints actually define the control points for the convex hull of the B-spline, but
the first and last waypoint are still met.
Dependencies
To enable this input, set Waypoint Source to
External
.
TimePoints — Time points for waypoints of trajectory
p-element vector
Time points for waypoints of trajectory, specified as a p-element vector.
Dependencies
To enable this input, set Waypoint Source to
External
.
VelBC — Velocity boundary conditions for waypoints
n-by-p matrix
Velocity boundary conditions for waypoints, specified as an n-by-p matrix. Each row corresponds to the velocity at each of the p waypoints for the respective variable in the trajectory.
Dependencies
To enable this input, set Method to Cubic
Polynomial
or Quintic Polynomial
and
Parameter Source to External
.
AccelBC — Acceleration boundary conditions for trajectory
n-by-p matrix
Acceleration boundary conditions for waypoints, specified as an n-by-p matrix. Each row corresponds to the acceleration at each of the p waypoints for the respective variable in the trajectory.
Dependencies
To enable this parameter, set Method to Quintic
Polynomial
and Parameter Source to
External
.
Output
q — Position of trajectory
scalar | vector | matrix
Position of the trajectory, specified as a scalar, vector, or matrix. If you specify a scalar for the Time input with an n-dimensional trajectory, the output is a vector with n elements. If you specify a vector of m elements for the Time input, the output is an n-by-m matrix.
Data Types: single
| double
qd — Velocity of trajectory
scalar | vector | matrix
Velocity of the trajectory, specified as a scalar, vector, or matrix. If you specify a scalar for the Time input with an n-dimensional trajectory, the output is a vector with n elements. If you specify a vector of m elements for the Time input, the output is an n-by-m matrix.
Data Types: single
| double
qdd — Acceleration of trajectory
scalar | vector | matrix
Acceleration of the trajectory, specified as a scalar, vector, or matrix. If you specify a scalar for the Time input with an n-dimensional trajectory, the output is a vector with n elements. If you specify a vector of m elements for the Time input, the output is an n-by-m matrix.
Data Types: single
| double
Parameters
Waypoint source — Source for waypoints
Internal
(default) | External
Specify External
to specify the Waypoints
and Time points parameters as block inputs instead of block
parameters.
Waypoints — Waypoint positions along trajectory
n-by-p matrix
Positions of waypoints of the trajectory at given time points, specified as an
n-by-p matrix, where n is the
dimension of the trajectory and p is the number of waypoints. If you
specify the Method as B-spline
, these waypoints
actually define the control points for the convex hull of the B-spline, but the first
and last waypoint are still met.
Dependencies
To specify this parameter in the block mask, set Waypoint
Source to Internal
.
Time points — Time points for waypoints of trajectory
p-element vector
Time points for waypoints of trajectory, specified as a p-element vector, where p is the number of waypoints.
Dependencies
To specify this parameter in the block mask, set Waypoint
Source to Internal
.
Method — Method for trajectory generation
Cubic Polynomial
(default) | Quintic Polynomial
| B-Spline
Method for trajectory generation, specified as either Cubic
Polynomial
, Quintic Polynomial
, or
B-Spline
.
Parameter source — Source for waypoints
Internal
(default) | External
Specify External
to specify the Velocity boundary
conditions and Acceleration boundary conditions
parameters as block inputs instead of block parameters.
Velocity boundary conditions — Velocity boundary conditions for waypoints
zeroes(2,5)
(default) | n-by-p matrix
Velocity boundary conditions for waypoints, specified as an n-by-p matrix. Each row corresponds to the velocity at each of the p waypoints for the respective variable in the trajectory.
Dependencies
To enable this input, set Method to Cubic
Polynomial
or Quintic Polynomial
.
Acceleration boundary conditions — Acceleration boundary conditions for trajectory
n-by-p matrix
Acceleration boundary conditions for waypoints, specified as an n-by-p matrix. Each row corresponds to the acceleration at each of the p waypoints for the respective variable in the trajectory.
Dependencies
To enable this parameter, set Method to Quintic
Polynomial
.
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 thanCode 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 toInterpreted execution
.
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] Farin, Gerald E. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. San Diego, CA: Academic Press, 1993.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2019a
See Also
Blocks
Functions
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