Trajectory Generation

A trajectory is a time-parameterized motion:

\[ x=x(t). \]

Moro 0.5.0 provides compact numerical point-to-point trajectories using polynomial interpolation.

Normalized time

Let

\[ T=t_f-t_0>0, \]

and define

\[ \tau=\frac{t-t_0}{T}. \]

Then

\[ 0\le \tau\le1. \]

Using normalized time makes the polynomial coefficients independent of the absolute time origin.

Physical-time derivatives satisfy

\[ \frac{d}{dt} = \frac{1}{T}\frac{d}{d\tau}, \]

and

\[ \frac{d^2}{dt^2} = \frac{1}{T^2}\frac{d^2}{d\tau^2}. \]

Linear interpolation

For endpoint positions \(x_0\) and \(x_f\),

\[ x(\tau)=x_0+(x_f-x_0)\tau. \]

Therefore

\[ \dot x=\frac{x_f-x_0}{T}, \]

and

\[ \ddot x=0. \]

Cubic interpolation

For

\[ x(\tau)=c_0+c_1\tau+c_2\tau^2+c_3\tau^3, \]

with endpoint velocities \(v_0\) and \(v_f\),

\[ c_0=x_0, \]

\[ c_1=Tv_0, \]

\[ c_2=3(x_f-x_0)-T(2v_0+v_f), \]

\[ c_3=-2(x_f-x_0)+T(v_0+v_f). \]

Velocity and acceleration are

\[ \dot x= \frac{1}{T} \left( c_1+2c_2\tau+3c_3\tau^2 \right), \]

\[ \ddot x= \frac{1}{T^2} \left( 2c_2+6c_3\tau \right). \]

Quintic interpolation

For

\[ x(\tau)= c_0+c_1\tau+c_2\tau^2+c_3\tau^3+c_4\tau^4+c_5\tau^5, \]

with endpoint velocities \(v_0,v_f\) and accelerations \(a_0,a_f\),

\[ c_0=x_0, \qquad c_1=Tv_0, \qquad c_2=\frac{T^2}{2}a_0. \]

Let

\[ \Delta x=x_f-x_0. \]

Then

\[ c_3= 10\Delta x -6Tv_0 -4Tv_f -\frac{3}{2}T^2a_0 +\frac{1}{2}T^2a_f, \]

\[ c_4= -15\Delta x +8Tv_0 +7Tv_f +\frac{3}{2}T^2a_0 -T^2a_f, \]

\[ c_5= 6\Delta x -3Tv_0 -3Tv_f -\frac{1}{2}T^2a_0 +\frac{1}{2}T^2a_f. \]

Classical zero-derivative profiles

For \(x_0=0\), \(x_f=1\), and zero supported derivative conditions:

Linear

\[ s(\tau)=\tau. \]

Cubic

\[ s(\tau)=3\tau^2-2\tau^3. \]

Quintic

\[ s(\tau)=10\tau^3-15\tau^4+6\tau^5. \]

This gives the teaching progression

linear  -> position
cubic   -> position + velocity
quintic -> position + velocity + acceleration

Vector trajectories

The same formulas apply componentwise to vector-valued coordinates.

For joint space,

\[ q,\dot q,\ddot q\in\mathbb R^{N\times n}. \]

For Cartesian position space,

\[ p,v,a\in\mathbb R^{N\times3}. \]

With zero derivative boundary conditions, a Cartesian position trajectory lies on the line segment between \(p_0\) and \(p_f\).

Nonzero derivative conditions can produce a curved path because the polynomial acts independently on each Cartesian component.

Separation from robot algorithms

Trajectory generation is independent of robot kinematics.

A Cartesian trajectory may later be resolved using sequential inverse kinematics:

cart = position_trajectory(...)
ik = solve_position_trajectory(robot, cart.p, q0=...)

Similarly, a joint-space trajectory can be passed directly to visualization as a time-major matrix.

No robot constraints or dynamic feasibility conditions are enforced by the trajectory generator itself.