Excitation Trajectory

Persistent excitation is the fundamental property that an input signal must possess to enable the consistent identification of all system parameters. A trajectory is persistently exciting if it provides sufficient stimulation across all dynamic modes and frequencies of interest over the entire data collection period. This ensures the regression matrix in the identification algorithm remains well-conditioned, preventing numerical ill-posedness and guaranteeing that all unknown parameters (e.g., inertia, friction coefficients) are uniquely identifiable from the collected data.

The trajectory must contain a broad frequency spectrum to excite the system's natural dynamics. This is often achieved by designing signals that are sums of sinusoids or via chirp signals that sweep through a target frequency band.

• Low-frequency content is needed to identify parameters related to gravity and static friction.
• Mid-to-high-frequency content is required to identify inertial and Coriolis effects.
• The bandwidth of the excitation must exceed the closed-loop bandwidth expected during normal operation to ensure the identified model is valid for subsequent controller design.

The signal must have sufficient amplitude to produce measurable outputs above the noise floor of the sensors, ensuring a high signal-to-noise ratio. However, amplitudes must remain within the safe operational limits of the hardware to avoid damage. The duration must be long enough to capture transient responses and allow for the decay of initial condition effects, providing multiple cycles of the lowest frequency components. For a robotic arm, this often means the trajectory should move all joints through a significant portion of their workspace.

This involves designing the trajectory to maximize the Fisher information matrix or minimize the covariance of the parameter estimates. The goal is to make the estimation problem as well-conditioned as possible. Common optimality criteria include:

• D-optimality: Maximizes the determinant of the information matrix, minimizing the volume of the confidence ellipsoid for the parameters.
• A-optimality: Minimizes the trace of the parameter covariance matrix.
• E-optimality: Maximizes the minimum eigenvalue of the information matrix. These criteria often result in trajectories that appear random or pseudo-random but are mathematically optimized for precision.

The designed trajectory must respect all actuator limits (torque, velocity, acceleration) and joint limits of the physical robot. It must avoid configurations that lead to singularities or self-collision. For safety, the trajectory is often executed under high-gain position control or in a gravity-compensation mode before moving to torque control for data collection. This characteristic ensures the experiment is repeatable and does not damage the hardware, which is critical for industrial and research applications.

Several standard trajectory classes are used in practice:

• Sum-of-Sinusoids: A linear combination of sine waves at different frequencies, allowing precise control over spectral content.
• Pseudo-Random Binary Sequence (PRBS): A signal that switches between two levels in a random but repeatable pattern, providing wideband excitation.
• Chirp Signal: A sine wave with a frequency that increases or decreases linearly over time, efficiently sweeping a frequency band.
• Optimal Robotic Excitations: Trajectories parameterized by Fourier series or polynomial coefficients, where the coefficients are optimized directly for a system identification criterion (e.g., D-optimality).

A swept-sine or chirp signal is a sinusoidal input whose frequency increases (or decreases) linearly or logarithmically over time. This technique is foundational for frequency-domain system identification.

• Primary Use: Excites a broad, continuous band of frequencies to characterize the system's frequency response and resonant modes.
• Key Property: Provides excellent persistent excitation across the specified bandwidth, ensuring all relevant dynamic modes are stimulated.
• Implementation: Defined as (u(t) = A \sin(2\pi (f_0 + kt) t)), where (f_0) is the start frequency and (k) is the sweep rate. Logarithmic sweeps are often used to ensure equal energy per octave.
• Advantage: Directly reveals system bandwidth and phase margins. It is a standard signal for identifying transfer functions in linear time-invariant systems.

A Pseudo-Random Binary Sequence is a deterministic, noise-like signal that switches between two predefined amplitude levels according to a maximum-length shift register sequence.

• Primary Use: Excites a wide spectrum of frequencies with a signal that has well-defined statistical properties and is safe for amplitude-limited actuators.
• Key Property: Its autocorrelation function approximates a Dirac delta function, making it spectrally similar to white noise but with bounded amplitude.
• Implementation: The sequence length (L = 2^n - 1) (where (n) is the register order) determines the frequency resolution. The clock period sets the highest excited frequency.
• Advantage: Maximizes signal power within actuator saturation limits, providing high signal-to-noise ratio for parameter estimation while maintaining safe operating conditions.

A multi-sine signal is a sum of sinusoids at specific, non-harmonically related frequencies, with carefully chosen phases and amplitudes.

• Primary Use: Provides precise control over which frequencies are excited, allowing targeted probing of the system's dynamics at points of interest.
• Key Property: Enforces persistent excitation at a discrete set of frequencies. Phase optimization (e.g., using Schroeder phases) minimizes the crest factor (peak-to-RMS ratio), distributing energy efficiently.
• Implementation: (u(t) = \sum_{k=1}^{N} A_k \sin(2\pi f_k t + \phi_k)). The frequencies (f_k) are chosen based on prior knowledge of the system's bandwidth.
• Advantage: Enables non-parametric frequency response estimation at the chosen points with high accuracy and allows for the separation of linear and nonlinear dynamics in the response.

Optimal experiment design formulates the trajectory generation as an optimization problem to maximize the information content in the collected data for parameter estimation.

• Primary Use: To design trajectories that minimize the statistical uncertainty (covariance) of the estimated parameters, subject to physical constraints.
• Key Objective: Maximize a criterion based on the Fisher Information Matrix (FIM), such as D-optimality (maximizing the determinant of the FIM) or E-optimality (maximizing its minimum eigenvalue).
• Constraints: The optimization incorporates actuator torque/velocity limits, joint position limits, and collision avoidance to ensure the trajectory is executable and safe.
• Advantage: Produces the most informative data possible within the system's operational envelope, leading to the highest possible parameter estimation accuracy for a given experiment duration.

This technique designs trajectories that ensure the dynamic regressor matrix—which linearly maps parameters to torque—is well-conditioned for the specific parameters of interest.

• Primary Use: Directly targets the identifiability of inertial and friction parameters in rigid-body dynamics models, such as the Newton-Euler or Lagrangian formulations.
• Core Principle: The robot's equations of motion can be written linearly in the dynamic parameters: (\tau = Y(q, \dot{q}, \ddot{q}) \pi), where (Y) is the regressor and (\pi) is the parameter vector. The trajectory is designed to make (Y) full rank.
• Method: Often involves solving an optimization to maximize the minimum singular value of the aggregated regressor matrix over the trajectory, ensuring all columns of (Y) are independently excited.
• Advantage: Provides guarantees on parameter estimation convergence for the specific dynamic model and is the standard method for high-precision inertial parameter identification in robotics.

Task-space excitation designs motions in the robot's operational space (e.g., Cartesian end-effector paths) that indirectly create rich joint-space excitation.

• Primary Use: To identify parameters relevant to task performance (like tool dynamics) or when joint-space trajectories are constrained by the workspace.
• Method: Defines excitation paths like figure-eights, Lissajous curves, or random walks in Cartesian space. The inverse kinematics solution generates the corresponding joint trajectory.
• Consideration: Must account for kinematic singularities and the non-linear mapping from task to joint space, which can affect the persistence of excitation.
• Advantage: Directly excites the dynamics in the coordinate frame where the robot is controlled and performs its primary function, aligning the identification experiment with the ultimate use case.

The process of constructing mathematical models of dynamic systems from measured input-output data. It involves:

• Experiment Design: Creating inputs, like excitation trajectories, to elicit informative outputs.
• Model Selection: Choosing a model structure (e.g., linear, nonlinear, grey-box).
• Parameter Estimation: Using algorithms to fit the model's unknown constants to the data.
• Validation: Testing the model's predictive power on a separate dataset. The goal is to characterize a system's behavior, enabling accurate simulation and control design.

A critical property of an input signal required for reliable system identification. A signal is persistently exciting if it contains enough frequency content and variation over time to stimulate all the dynamic modes of the system. An excitation trajectory must be designed for persistent excitation to ensure that:

• All unknown parameters can be uniquely estimated.
• The estimation algorithms converge to accurate values.
• The identified model is not biased by insufficient data. Without this property, some system dynamics remain unobserved and unmodeled.

The computational process of inferring the unknown constant values within a system's mathematical model from observed data. For a robotic arm, these parameters include link masses, inertia tensors, and friction coefficients. Methods include:

• Least Squares: Fits parameters by minimizing the squared error between model predictions and measured data.
• Maximum Likelihood Estimation: Finds parameters that make the observed data most probable.
• Bayesian Inference: Treats parameters as probability distributions updated by data. Excitation trajectories provide the rich data required for these algorithms to function accurately.

A key mathematical formulation that enables linear parameter estimation for complex robotic systems. It rewrites the nonlinear equations of motion into a linear form: Y(q, q̇, q̈) * Φ = τ, where:

• Y is the regressor matrix, composed of known functions of joint positions, velocities, and accelerations.
• Φ is the vector of unknown dynamic parameters (e.g., inertial and frictional terms).
• τ is the vector of measured joint torques. The excitation trajectory is designed to make the regressor matrix Y well-conditioned, allowing for the robust numerical solution for Φ.

The degree to which a simulation model accurately replicates the behavior and outputs of the real-world system. It is the ultimate goal of using excitation trajectories for system identification. High-fidelity models exhibit:

• Kinematic Accuracy: Correct geometric and positional relationships.
• Dynamic Accuracy: Correct forces, torques, and accelerations for given inputs.
• Predictive Validity: The ability to forecast system behavior under new conditions. Low-fidelity models, resulting from poor identification, lead to a large reality gap, degrading the performance of simulation-trained policies.

A standardized, repeatable procedure for performing system identification on a specific class of systems (e.g., 6-DOF robotic arms, quadrupeds). A robust protocol defines:

1. Excitation Design: Specifications for the trajectory (e.g., frequency bandwidth, amplitude).
2. Data Collection: Sensor setup, sampling rates, and filtering.
3. Pre-processing: Steps for calculating velocities/accelerations and synchronizing data.
4. Estimation & Validation: The specific algorithms to run and the metrics for success. Protocols ensure results are consistent, comparable, and that the identified model is suitable for its intended use, such as high-fidelity simulation.