Trapezoidal Velocity Profile — Robotics/Trajectory Generation

Trapezoidal Velocity Profile

Robotics / Trajectory Generation

The bounded-velocity point-to-point profile: a source that walks from q₀ to q_f as fast as a velocity limit and an acceleration limit allow, in three segments – accelerate, cruise, decelerate – and then holds. It is what a motion controller commands when the machine, not the move time, sets the pace.

With D = |q_f − q₀|, the peak velocity actually reached is v = min(v_max, √(D·a_max)), and the segment times follow: t_a = v/a_max, t_c = (D − v·t_a)/v, T = 2t_a + t_c.

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 – the velocity command, [1,1]. This is the trapezoid the block is named for: a ramp up, a flat top at ±v, a ramp down.
• qdd – the acceleration command, [1,1]. It is ±a_max, then 0, then ∓a_max, then 0 – piecewise constant, and it steps.

Parameters

• Start Position – q₀, held before the move and at t = 0.
• End Position – q_f, reached at t = T and held after it. It may be below q₀; the profile runs downhill and every output changes sign with it.
• Max Velocity – v_max, the flat top of the trapezoid. It must be strictly positive. If the move is too short to reach it the profile becomes a triangle – see the notes.
• Max Acceleration – a_max, the slope of the ramps. It must be strictly positive.
• 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.

All four parameters are baked into the emitted body at export time rather than exposed as tunable parameters on the generated core. The segment times, the peak velocity and the signed acceleration are solved once and enter as constants, so no generated core divides, takes a square root, or searches for the segment it is in – it compares against two or three numbers. 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 arithmetic itself would be synthesizable fixed point – it is comparisons and multiply-add – but the clock is not: like every time-driven source in this library it reads the testbench-advanced sim_time.

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 four segment cases 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 triangle case is not an error. When the move is too short to reach v_max – that is, when D·a_max ≤ v_max² – the cruise segment vanishes, the peak velocity is √(D·a_max) instead of v_max, and the velocity command is a triangle rather than a trapezoid. Both limits are still respected.
• The deceleration segment is computed backwards from T, as the mirror of the acceleration one, so the profile arrives at q_f exactly rather than at the sum of three roundings.
• q̈ steps. Nothing bounds jerk here, so the acceleration command jumps at each segment boundary. Use S Curve Profile where that matters, or Quintic Trajectory where the move TIME is what you want to set rather than the limits.
• A zero-length move is not a special case: with q_f = q₀ the duration is zero and the block holds (q₀, 0, 0) from the first sample.
• No state space – the block has no input to be linear in.