Quintic Trajectory — Robotics/Trajectory Generation

Quintic Trajectory

Robotics / Trajectory Generation

The minimum-jerk point-to-point profile: a source that walks from q₀ to q_f over a duration T, starting and finishing at zero velocity and zero acceleration. With normalized time s = t/T clamped to [0, 1]:

• q = q₀ + h·(10s³ − 15s⁴ + 6s⁵), where h = q_f − q₀
• q̇ = (h/T)·(30s² − 60s³ + 30s⁴)
• q̈ = (h/T²)·(60s − 180s² + 120s³)

The quintic is the lowest-order polynomial that can meet all six boundary conditions (position, velocity and acceleration at each end), which is why it is what a servo is actually commanded with rather than a straight ramp.

Ports

This block is a source: it has no inputs and takes its time from its own local clock, so it produces the same profile wherever it is dropped.

• q – the position command, [1,1].
• qd – its first derivative, the velocity command, [1,1].
• qdd – its second derivative, the acceleration command, [1,1].

Parameters

• Start Position – q₀, the value held before the move and at t = 0. Any scalar.
• End Position – q_f, the value reached at t = T and held after it. It may be below q₀; the profile simply runs downhill.
• Duration (s) – T, the move time. It must be strictly positive; a zero or negative value is reported and stops the run rather than dividing.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text.

q₀, q_f and T are baked into the emitted body at export time rather than exposed as tunable parameters on the generated core. They enter as the folded constants q₀, h, 1/T, h/T and h/T², so no generated core performs a division – and retuning the profile means re-exporting it.

The three HDL targets are simulation-only: they carry the profile in real and quantize only at the port boundary. The polynomial itself would be synthesizable fixed point – it is multiply-add on constants with every power inside [0, 1] – but the clock is not: like every time-driven source in this library it reads the testbench-advanced sim_time, and there is no fixed-point clock to read instead.

Simulink bridge

None, measured rather than assumed: point-to-point trajectory blocks ship in the Robotics System Toolbox, which is not installed on this machine (robotics, nav and shared_robotics are all absent from matlabroot/toolbox). The profile is easy to rebuild there from a Clock and a MATLAB Function block carrying the three polynomials above.

Notes

• Algebraic and stateless – a pure function of simulation time, so it runs correctly under either solver and carries no state to seed.
• The hold after t > T is the clamp, not a separate branch: at s = 1 the three polynomials are exactly 1, 0 and 0, so the block publishes (q_f, 0, 0) from then on as an identity of the arithmetic. Before t = 0 – which only a negative model start time can reach – it holds (q₀, 0, 0) the same way.
• The profile is evaluated in normalized time, which is what keeps it exact in fixed point: in raw t the acceleration term carries t³ and would leave Q16.16's range within seconds.
• No state space – the block has no input to be linear in.
• For a bounded-velocity move use Trapezoidal Velocity Profile: this one's peak speed is 1.875·h/T and its peak acceleration 5.7735·h/T², both set by T rather than commanded.