S Curve Profile — Robotics/Trajectory Generation

S Curve Profile

Robotics / Trajectory Generation

The jerk-limited point-to-point profile – the seven-segment or “double-S” shape a production motion controller commands. It is Trapezoidal Velocity Profile with a jerk ramp spliced into every corner, so the acceleration command is continuous rather than stepping, and the machine is not shocked at each segment boundary.

1. jerk +j – acceleration ramps from 0 to its peak.
2. jerk 0 – acceleration held at its peak.
3. jerk −j – acceleration ramps back to 0; velocity arrives at its peak.
4. jerk 0 – cruise.
5. jerk −j, 6. jerk 0, 7. jerk +j – the mirror image, decelerating to rest.

Ports

This block is a source: it has no inputs and takes its time from its own local clock.

• q – the position command, [1,1].
• qd – the velocity command, [1,1]. An S rather than a trapezoid: its corners are rounded by the jerk ramps.
• qdd – the acceleration command, [1,1]. Piecewise linear and continuous, which is the whole point of the block.

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.
• Max Velocity – v_max, strictly positive.
• Max Acceleration – a_max, strictly positive.
• Max Jerk – j_max, strictly positive. This is what separates this block from Trapezoidal Velocity Profile: as it grows the profile approaches that block's, and as it shrinks the move stretches and softens.
• 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 five parameters are baked into the emitted body at export time rather than exposed as tunable parameters. The whole profile – the shape, the segment boundaries and each segment's starting state – is solved once and enters as constants, so no generated core takes a square root, a cube root or a division. Each segment is one cubic in the time since that segment began. 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 would be synthesizable fixed point; the clock is not, since like every time-driven source here it reads the testbench-advanced sim_time.

Simulink bridge

None, measured rather than assumed: jerk-limited 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 can be rebuilt there from a Clock and a MATLAB Function block carrying the segment cases above.

Notes

• Algebraic and stateless – a pure function of simulation time.
• The move has four possible shapes, and all four are normal. Whether a_max is reached and whether v_max is reached are independent questions: reaching both gives the full seven segments; reaching only the acceleration limit drops the cruise; reaching only the velocity limit drops the two constant-acceleration segments; reaching neither leaves four jerk ramps and nothing else. Every limit is respected in all four.
• The peak values are reported by the profile, not assumed. On a short move the velocity peaks below v_max and the acceleration may peak below a_max; that is the profile obeying the jerk limit, not a failure to reach the others.
• It arrives exactly. Each segment is evaluated from its own start state, so q(T) = q_f and q̇(T) = q̈(T) = 0 to within a rounding of the arithmetic rather than to within the sum of seven of them.
• 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.