Dynamic Movement Primitives (DMP)

A numerical optimization approach that refines an initial path into a dynamically feasible trajectory by minimizing a cost function subject to kinematic and collision constraints. While DMPs provide a stable attractor landscape for motion, trajectory optimization techniques like direct collocation or shooting methods can be used to compute the optimal forcing term that shapes the DMP's nonlinear dynamics, combining the flexibility of optimization with the inherent stability guarantees of the dynamical system.

The computational process of determining joint parameters that achieve a desired end-effector pose, often solved using numerical methods like the Jacobian pseudoinverse. DMPs typically encode motion in task space (end-effector coordinates), requiring IK to map the generated trajectory into joint commands. The stability properties of DMPs are preserved through this mapping, ensuring that convergence to the goal in task space translates to a stable joint configuration, even when analytical IK solutions are unavailable.

A real-time control strategy that solves a finite-horizon optimization problem at each timestep to generate control inputs while respecting system dynamics and constraints. DMPs and MPC are complementary: a DMP can generate a nominal trajectory that serves as the reference signal for an MPC controller, which then handles online disturbance rejection and constraint enforcement. This hybrid approach leverages DMPs for rapid motion generalization and MPC for robust execution under uncertainty.

The mathematical space representing all possible positions and orientations of a robot, where path planning transforms into finding a continuous curve for a point. DMPs operate in the task space or joint space subset of C-Space. When generalizing a demonstrated trajectory to a new goal, the DMP's transformation properties ensure the adapted path remains within the collision-free subset of C-Space if the original demonstration was collision-free and the environment is sufficiently similar.

Motion planning that simultaneously considers kinematic constraints and differential dynamics, ensuring trajectories respect velocity, acceleration, and force limits. Standard DMP formulations can be extended to incorporate velocity and acceleration bounds by modulating the canonical system's temporal scaling or by reshaping the forcing term. This allows learned motor primitives to respect the robot's physical actuation limits without sacrificing the stability guarantees of the underlying dynamical system.