Trajectory Generation

Textbook Chapter 9

ModernRoboticsBook.cartesian_trajectory — Method
cartesian_trajectory(transform_start, transform_end, total_time, N, method)

Computes a trajectory as a list of $N$ SE(3) matrices where the origin of the end-effector frame follows a straight line and the rotation follows a geodesic on SO(3). Unlike screw_trajectory, the position is decoupled from the rotation.

Screw vs Cartesian trajectory

Unlike screw_trajectory, a Cartesian trajectory decouples rotation and translation: the origin follows a straight line in space while the orientation interpolates independently along the shortest path (geodesic) on SO(3). This is usually more intuitive for pick-and-place tasks.

Arguments

  • transform_start: the initial end-effector configuration (4×4 SE(3) matrix).
  • transform_end: the final end-effector configuration (4×4 SE(3) matrix).
  • total_time: total time of the motion in seconds from rest to rest.
  • N: the number of points $N \geq 2$ in the discrete representation of the trajectory.
  • method: the time-scaling method — use 3 for cubic (cubic_time_scaling) or 5 for quintic (quintic_time_scaling).

Returns

A list of $N$ SE(3) matrices separated in time by $T_f / (N - 1)$. The first in the list is transform_start and the $N$th is transform_end.

Examples

julia> Xstart = [1 0 0 1; 0 1 0 0; 0 0 1 1; 0 0 0 1];

julia> Xend = [0 0 1 0.1; 1 0 0 0; 0 1 0 4.1; 0 0 0 1];

julia> traj = cartesian_trajectory(Xstart, Xend, 5, 4, 5);

julia> length(traj)
4

julia> traj[1]
4×4 Matrix{Float64}:
 1.0  0.0  0.0  1.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  1.0
 0.0  0.0  0.0  1.0
source
ModernRoboticsBook.cubic_time_scaling — Method
cubic_time_scaling(total_time, t)

Computes $s(t) = 3(t/T_f)^2 - 2(t/T_f)^3$ for a cubic time scaling, satisfying $s(0)=0$, $s(T_f)=1$, $\dot{s}(0)=0$, $\dot{s}(T_f)=0$.

Arguments

  • total_time: the total time $T_f$ of the motion in seconds.
  • t: the current time $t$ satisfying $0 \le t \le T_f$.

Returns

A scalar $s \in [0, 1]$ representing the fraction of motion completed.

Examples

julia> cubic_time_scaling(2, 0.6)
0.21600000000000003
source
ModernRoboticsBook.joint_trajectory — Method
joint_trajectory(joint_position_start, joint_position_end, total_time, N, method)

Computes a straight-line trajectory in joint space with the specified time scaling.

Arguments

  • joint_position_start: the $n$-vector of initial joint variables.
  • joint_position_end: the $n$-vector of final joint variables.
  • total_time: total time of the motion in seconds from rest to rest.
  • N: the number of points $N \geq 2$ in the discrete representation of the trajectory.
  • method: the time-scaling method — use 3 for cubic (cubic_time_scaling) or 5 for quintic (quintic_time_scaling).

Returns

An $N×n$ matrix where each row is an $n$-vector of joint variables. The first row is joint_position_start and the $N$th row is joint_position_end. The elapsed time between each row is $T_f / (N - 1)$.

Examples

julia> joint_trajectory([0, 0], [1, 2], 2, 4, 3)
4×2 adjoint(::Matrix{Float64}) with eltype Float64:
 0.0       0.0
 0.259259  0.518519
 0.740741  1.48148
 1.0       2.0
source
ModernRoboticsBook.quintic_time_scaling — Method
quintic_time_scaling(total_time, t)

Computes $s(t) = 10(t/T_f)^3 - 15(t/T_f)^4 + 6(t/T_f)^5$ for a quintic time scaling, satisfying $s(0)=0$, $s(T_f)=1$, $\dot{s}(0)=0$, $\dot{s}(T_f)=0$, $\ddot{s}(0)=0$, $\ddot{s}(T_f)=0$.

Arguments

  • total_time: the total time $T_f$ of the motion in seconds.
  • t: the current time $t$ satisfying $0 \le t \le T_f$.

Returns

A scalar $s \in [0, 1]$ representing the fraction of motion completed.

Examples

julia> quintic_time_scaling(2, 0.6)
0.16308
source
ModernRoboticsBook.screw_trajectory — Method
screw_trajectory(transform_start, transform_end, total_time, N, method)

Computes a trajectory as a list of $N$ SE(3) matrices corresponding to the screw motion about a space screw axis. Unlike cartesian_trajectory, the origin follows a screw path rather than a straight line.

What is a screw trajectory?

A screw trajectory moves the end-effector along a constant screw axis — like turning a corkscrew. The rotation and translation are coupled: the end-effector follows a helical path. Compare with cartesian_trajectory, which decouples them.

Arguments

  • transform_start: the initial end-effector configuration (4×4 SE(3) matrix).
  • transform_end: the final end-effector configuration (4×4 SE(3) matrix).
  • total_time: total time of the motion in seconds from rest to rest.
  • N: the number of points $N \geq 2$ in the discrete representation of the trajectory.
  • method: the time-scaling method — use 3 for cubic (cubic_time_scaling) or 5 for quintic (quintic_time_scaling).

Returns

A list of $N$ SE(3) matrices separated in time by $T_f / (N - 1)$. The first in the list is transform_start and the $N$th is transform_end.

Examples

julia> Xstart = [1 0 0 1; 0 1 0 0; 0 0 1 1; 0 0 0 1];

julia> Xend = [0 0 1 0.1; 1 0 0 0; 0 1 0 4.1; 0 0 0 1];

julia> traj = screw_trajectory(Xstart, Xend, 5, 4, 3);

julia> length(traj)
4

julia> traj[1]
4×4 Matrix{Float64}:
 1.0  0.0  0.0  1.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  1.0
 0.0  0.0  0.0  1.0
source