Double Integrator
- Category: Control › Feedback control
- Tags: dynamics, linear-systems, state-space, control-theory, point-mass
The simplest second-order dynamic system: the control input is acceleration, and position and velocity are the states. It models a point mass moving under a commanded force.
Definition
A double integrator is a system whose input is integrated twice to give its output. In one dimension the dynamics are ẍ = u: the control u is an acceleration, velocity is its integral and position is the integral of velocity. In state-space form, with state [p; v], the system is ẋ = A x + B u with A = [[0, 1], [0, 0]] and B = [0; 1]. Higher-dimensional versions stack one such pair per axis, so a point mass moving in 3D is three independent double integrators. Physically it describes a frictionless mass driven by a force, divided by the mass.
The double integrator is linear and controllable, which makes it a standard example in control and planning. Proportional-derivative (PD) feedback on position and velocity stabilizes it, and the gains map directly to the natural frequency and damping of the closed loop. A linear-quadratic regulator for it comes from a small Riccati equation, and the minimum-time problem with bounded acceleration has a well-known bang-bang solution with a single switching curve in the phase plane. In discrete time with step Δt and constant input over the step, the update is p ← p + v Δt + ½ u Δt² and v ← v + u Δt, which is easy to embed in model predictive control and trajectory optimization.
Many real systems reduce to double integrators after a transformation. Computed-torque control cancels a fully actuated manipulator's nonlinear dynamics so that each joint behaves like a double integrator driven by a commanded acceleration. A quadrotor's translational motion is close to a double integrator in the commanded acceleration, which is one reason differential flatness and minimum-snap trajectories work well for drones. Planning papers often use double-integrator robots in simulation because they capture momentum and acceleration limits while keeping the dynamics simple. The cost of that simplicity is that friction, actuator dynamics, nonholonomic constraints and saturation are missing, so results on a double integrator do not automatically transfer to a real platform.
Why it matters
The double integrator is the simplest model that captures inertia: a robot cannot change its velocity instantly. It is the starting point for understanding feedback control, trajectory generation and kinodynamic planning, and many practical controllers are designed by first reducing a robot to this form.
Key applications
- Benchmark dynamics for motion planning and control research
- Simplified models of quadrotor translation and point-mass vehicles
- Joint-level models after computed-torque linearization
- Teaching PD, LQR and minimum-time control
- Multi-agent consensus and formation control with second-order agents
In practice
A team prototyping a reactive planner for a team of ground robots first tests it on simulated double-integrator agents with bounded acceleration and velocity, so that the planner has to account for stopping distances. Once the planner works, they deploy it on real differential-drive robots, using a low-level controller that tracks the planned accelerations and a margin on the acceleration limits to absorb tracking error. When the real robots overshoot turns, the team traces the problem to the unmodeled nonholonomic constraint and either adds it to the planner or tightens the acceleration bounds.
Common misconceptions
- A double integrator is only a textbook toy — many controlled systems reduce to it after feedback linearization, and it is widely used in drone and manipulator control
- Results on a double integrator transfer directly to real robots — friction, actuator limits, delays and nonholonomic constraints are absent from the model and can change the behavior
- A double integrator is the same as a single integrator — a single integrator takes velocity as input and can change velocity instantly, while a double integrator takes acceleration and has momentum
Who should know this
- Control engineers and students
- Motion planning researchers testing new algorithms
- Drone and mobile robot developers
- Multi-robot systems researchers
Related technologies
- PD Control
- Computed-Torque Control
- Bang-Bang Control
- State-Space Models
- Model Predictive Control
Related terms
- Linear-Quadratic Regulator
- PID Controller
- Model Predictive Control
- Feedback Linearization
- Minimum-Snap Trajectory
- Kinodynamic Planning