Stanford AA274A Principles of Robotic Autonomy | Autumn 2019 | Trajectory Optimization

Differential equations are crucial for describing robot motion. They relate the derivative of an unknown function (like position over time) to the function itself, allowing us to model and predict the robot's movement dynamics. Solving these equations numerically provides the robot's state history.

Euler's method is a simple numerical technique for approximating solutions to differential equations. It approximates the derivative of a function at a point by using the function's value and the time step (Δt) to predict the next state, essentially discretizing the continuous motion into small, calculable steps.

Path feasibility focuses on finding *any* path that satisfies the robot's motion equations, control input limits, and state constraints (like avoiding obstacles). Optimal path control goes further by seeking the best path among all feasible ones, according to a defined cost function (e.g., minimum time, minimum energy).

A cost function quantifies the 'undesirability' of a system's behavior over time. It typically includes a running cost (integrated over time, e.g., control effort) and a final cost (evaluated at the end state). The goal of optimal control is to minimize this total cost.

Open-loop control relies on a pre-calculated control signal that depends only on time. Closed-loop control, however, continuously measures the system's state and adjusts the control signal based on the current state, making it more robust to disturbances and model inaccuracies.

Closed-loop control is preferred because real-world models are imperfect, and disturbances can cause deviations from the planned path. A closed-loop system can sense these deviations and actively correct them, ensuring the robot stays on track.

There are two main approaches: direct methods, which discretize the system and solve it as a non-linear optimization problem, and indirect methods, which derive necessary optimality conditions first and then solve those.

Differential flatness is a property of certain systems where a subset of state variables (the 'flat outputs') can be chosen such that all other states and control inputs can be expressed as functions of these outputs and their derivatives, without requiring integration. This significantly simplifies path planning and control, especially for systems like cars and quadrotors.

For differentially flat systems, you can plan a path in the space of the flat outputs (e.g., X-Y coordinates for a car) using simple interpolation (like polynomials). Then, a mapping function derived from the system's dynamics translates this flat output path into the actual state and control trajectories.

Jerk refers to the third derivative of position with respect to time (the derivative of acceleration). Minimizing jerk and higher-order derivatives (snap, crackle) in trajectory planning results in smoother, more controllable paths that require less control effort and are less likely to cause discomfort or damage.

Input constraints can be handled by techniques like time-scaling. The path is first generated without considering the constraint, and then the trajectory is re-timed (slowed down or sped up) until the control inputs (like speed) meet their limits. This ensures the path remains feasible within the operational constraints.