Dynamic Movement Primitives in Robotics

• Dynamic Movement Primitives are parametric nonlinear dynamical systems that encode, generate, and adapt complex robot motions using stable, goal-attractor dynamics.
• They decouple transformation (movement shape) from phase progression (timing) to enable robust and flexible adaptation to new goals and durations.
• Extensions such as geometric, probabilistic, and constraint-aware DMPs expand applications from industrial automation to human–robot interaction with enhanced safety and performance.

Dynamic Movement Primitives (DMPs) are a class of parametric, nonlinear dynamical systems designed for encoding, generating, and adapting complex robot motions. Originating from the motor primitives theory in biological sensorimotor control, DMPs provide a unified mathematical framework for learning from demonstration (LfD), generalizing motor skills, and robust online adaptation across varied robotics domains (Saveriano et al., 2021). They are implemented as stable dynamical systems with goal attractor properties, which can be efficiently learned, modulated, and sequenced to synthesize high-dimensional behaviors with strong guarantees on convergence, stability, and reactivity.

1. Mathematical Formulation and Representational Properties

In the standard discrete (point-to-point) case, a DMP is defined for each trajectory dimension by a second-order differential system:

$$\tau \dot{z} = \alpha_z [\beta_z (g - y) - z] + f(x)$$

$\tau \dot{y} = z$

$\tau \dot{x} = -\alpha_x x$

• $y$ is the system position, $g$ is the goal, $z$ is a scaled velocity, $\tau$ is a temporal scaling parameter.
• The phase variable $x$ monotonically decays from 1 to 0, ensuring time-invariant progression.
• The non-linear forcing term $f(x)$ has the form:

$$f(x) = \frac{\sum_{i=1}^N w_i \Psi_i(x)}{\sum_{i=1}^N \Psi_i(x)} x$$

$\Psi_i(x) = \exp(-h_i(x-c_i)^2)$

The weights $w_i$ are learnable parameters fitted to demonstrations using locally weighted regression or other algorithms.

DMPs ensure global convergence to $g$ and allow stability to be tuned via $\alpha_z$ and $\beta_z$ (typically $\alpha_z = 4\beta_z$). By separating the transformation system (behavior/shape) from the canonical system (phase/timing), DMPs can robustly adapt to new goals and durations (Saveriano et al., 2021).

For rhythmic behaviors, the DMP system is modified to produce limit cycles by replacing the point attractor with cyclic attractor dynamics. Multi-degree-of-freedom trajectories are handled through parallel DMPs with shared phase variables for temporal synchronization.

2. Learning from Demonstration and Skill Adaptation

DMPs are trained by recording demonstration trajectories and computing a corresponding time series of target forcing terms:

$$f_d(t_j) = \tau^2 \ddot{y}_d(t_j) - \alpha_z [\beta_z (g - y_d(t_j)) - \tau \dot{y}_d(t_j)]$$

Weights $w_i$ are solved by regularized least-squares regression over the basis functions $\Psi_i(x)$.

Key adaptation mechanisms include:

• Changing $g$ and $y_0$ at runtime for generalizing goal-directed motion.
• Modulating $\tau$ to adjust execution speed.
• Modifying phase progression for phase-stopping or goal switching:

$$\tau \dot{x} = -\frac{\alpha_x x}{1+\alpha_{yx} ||\tilde{y}-y||}$$

• Blending multiple primitives by overlaying basis functions for smooth transitions, or by sequencing with velocity-based switching.

DMPs are extensively integrated with higher-level motion planners and task representations (Behavior Trees, Temporal Logic, Task and Motion Planning frameworks), yielding robust, scalable, and interpretable motion synthesis pipelines (Liu et al., 19 Jul 2025, Wang et al., 2022).

3. Extensions for Geometric, Probabilistic, and Constraint-Aware Motion

To address limitations of the Euclidean formulation, DMPs have been extended to non-Euclidean manifolds:

• Quaternion and rotation matrix DMPs for orientation, using logarithmic and exponential maps to guarantee geometric constraints (Abu-Dakka et al., 2022, Abu-Dakka et al., 2021).
• Riemannian DMPs for SPD manifold data (e.g., stiffness, manipulability) via affine-invariant log/exp maps (Abu-Dakka et al., 2020, Abu-Dakka et al., 2022).
• Composite manifold DMPs for tasks coupling position, orientation, and impedance (Abu-Dakka et al., 2022).

Probabilistic DMPs reformulate the system as a controlled linear dynamical system with Gaussian process noise (typically, uncertainty in $f(x)$), enabling principled state estimation, Kalman filtering, and failure detection by likelihood monitoring (Meier et al., 2016).

DMPs can be augmented to guarantee non-holonomic (velocity-level) constraints by closed-form coupling terms derived from the Udwadia-Kalaba method, ensuring compliance in constrained manipulation such as cutting, steering, or insertion tasks (Straižys et al., 2022).

Residual learning approaches combine DMPs with RL-based residual policies in task space, significantly increasing robustness, adaptation, and safety for contact-rich or uncertain manipulation (Davchev et al., 2020).

4. Practical Applications and Performance in Robotics

DMPs have been validated in a wide range of robotic applications:

• Human-robot skill transfer and personalization: DMP parameters extracted from user demonstrations enable trajectories personalized to ergonomic or habitual user features, supporting collaborative manipulation and lowering cognitive/physiological load as measured by EDA and EEG (Franceschi et al., 11 Jun 2025).
• Expressive and social robotics: DMP modulation enables realization of animation principles, expressive or stylized motion in social robots, and compositional design of gesture libraries (Hielscher et al., 9 Apr 2025).
• Industrial automation: Geometry-aware extensions (MeshDMP) allow learning, generalization, and real-time adaptation of surface-following skills on complex meshes, demonstrated in car fender polishing and cleaning (Vedove et al., 19 Oct 2024).
• Obstacle avoidance, sequenced and logic-guided tasks: DMP weights can be optimized to satisfy temporal logic constraints for long-horizon tasks and to incorporate user preferences efficiently (Wang et al., 2022, Liu et al., 19 Jul 2025).
• Physical interaction and co-manipulation: Phase-independent (Geometric DMP) variants decouple spatial path from timing, facilitating human-in-the-loop, phase-reversible, and optimality-constrained execution with provable passivity and stability (Braglia et al., 16 Jan 2024).

Representative experimental and simulation results consistently show that DMP-based formulations improve adaptation, predictability, and safety across diverse robotic platforms and manipulation tasks.

5. Advantages, Limitations, and Theoretical Properties

Advantages

• DMPs guarantee global convergence to the goal, with strong stability properties and reactivity to online perturbations.
• Modular and composable: primitives can be sequenced, blended, or adapted for hierarchical skill encoding.
• Support for efficient learning, imitation, and generalization from limited demonstrations.
• Geometrically principled extensions cover a broad array of real robot skill needs (poses in Euclidean and manifold spaces, variable impedance, etc.).
• Probabilistic variants handle uncertainty, sensor fusion, and failure detection without ad-hoc feedback design.

Limitations

• Implicit time-dependence in phase variable can induce synchronization issues for divergent demonstration timings (Saveriano et al., 2021).
• Classical DMPs are deterministic; only probabilistic extensions represent skill variability (Meier et al., 2016).
• Single-attractor bias limits flexible multi-goal or multi-attractor skills unless extended architectures are employed.
• Conventional DMPs are not inherently geometry-aware, requiring Riemannian or manifold-specific formulations for non-Euclidean robot features (Abu-Dakka et al., 2022).
• Parameter sensitivity (e.g., start-goal mirroring, scaling artifacts) mitigated by affine-invariant formulations and suitable basis function design (Ginesi et al., 2019).

6. Research Directions and Current Frontiers

Contemporary research addresses:

• Fully unified geometry-aware DMPs for arbitrary Riemannian and composite manifolds (Abu-Dakka et al., 2022).
• Probabilistic movement primitives encompassing uncertainty both in skill encoding and during reproduction, supporting learning, execution, and failure monitoring (Meier et al., 2016).
• Integration of DMPs into logic-constrained, behavior-tree-based, and reinforcement learning frameworks for long-horizon, hierarchical, and context-aware manipulation (Liu et al., 19 Jul 2025, Wang et al., 2022).
• User-personalized and adaptive motion generation through human-in-the-loop, physiologically informed control, and ergonomic-infused primitives (Franceschi et al., 11 Jun 2025).
• Phase-independent skill representations and optimal time-scaling for co-manipulation and real-time physical human-robot interaction (Braglia et al., 16 Jan 2024).

Emerging extensions include compactly supported, mollifier-like basis functions for improved numerical properties (Ginesi et al., 2019), residual learning overlays for contact-rich or high-uncertainty settings (Davchev et al., 2020), and robust multi-demonstration regression methods (Ginesi et al., 2019).

7. Summary Table: DMP Properties and Supported Extensions

Feature	Standard DMPs	Geometry-aware DMPs	Probabilistic DMPs	Residual RL-DMPs
Space	$\mathbb{R}^n$	Arbitrary Riemannian manifold	$\mathbb{R}^n$, stochastic	$\mathbb{R}^n$ + task space
Goal adaptation	Yes	Yes (intrinsic to formulation)	Yes	Yes
Sequence/composition	Yes	Yes	Yes	Yes
Uncertainty handling	No	No	Yes	Yes (via RL)
Constraint satisfaction	No	Yes (in manifold)	Limited	Yes (task space RL)
Human-in-the-loop compatibility	Limited	High (with phase-independent)	No	Yes

DMPs establish a foundational formalism for skill encoding and adaptive motion generation in robotics. The landscape of current research reflects the continuing integration of geometric, probabilistic, and constraint-oriented perspectives, driven by demands for higher generality, robustness, and usability in complex, interactive, and collaborative environments.