Probabilistic Trajectory Learning

• Probabilistic trajectory learning is a framework that represents entire motion sequences as random variables, capturing natural variability and uncertainty.
• It integrates calibrated sensor data from low-cost devices by aligning human demonstration signals with robot states for robust reproduction.
• The approach supports adaptive teleoperation control by modulating feedback and controller gains based on time-varying covariance, enhancing task repeatability.

Probabilistic trajectory learning is an approach for modeling, imitation, and reproduction of motion sequences wherein entire trajectories are represented as random variables, allowing natural encoding of both variability and uncertainty. This formalism is particularly suited for teleoperated robotics, learning from demonstration, and sensor-integrated feedback systems, as exemplified by the low-cost sensor glove platform for bilateral robot teleoperation described by Kragic et al. (Rueckert et al., 2015). The combination of hardware-constrained input, expressive software interface, and trajectory-level probabilistic models enables reproducible, robust skill transfer under real-world conditions.

1. Conceptual Basis of Probabilistic Trajectory Representations

Probabilistic trajectory learning encodes each trajectory as a sequence of random vectors $y_{1:T}$, each drawn from a multivariate distribution typically parameterized by a mean function $\mu(t)$ and covariance function $\Sigma(t)$. This framework allows practitioners to capture the natural variability observed in multiple human demonstrations—rather than constraining the model to a single mean path—enabling smooth generalization, stochastic reproduction, and explicit quantification of uncertainty at each timestep.

In standard formalism, trajectories are modeled as samples from

$y_{1:T} \sim \mathcal{N}(\mu_{1:T}, \Sigma_{1:T})$

where $\mu_{1:T}$ and $\Sigma_{1:T}$ are time-varying empirical estimates computed from an aligned demonstration dataset.

2. Data Acquisition and Signal Mapping

To learn probabilistic trajectory models from human demonstrations using sensor gloves, a pipeline must ensure precise and synchronized capture of relevant signals:

• The presented hardware comprises five resistive bend sensors sampled at up to 350 Hz. The raw resistance values $R_i(t)$ (measured via a voltage divider, digitized by a 10-bit ADC) are mapped to joint angles

$\theta_i(t) = \frac{R_i(t) - R_{0,i}}{k_{R,i}}$

where $R_{0,i}$, $k_{R,i}$ are per-finger calibration parameters computed offline.

• The glove angles are time-aligned with external motion capture to generate full teleoperation state vectors $y_t$ that include finger joints and pose information.

The data acquisition loop proceeds as:

1. Read sensor signals at high frequency (up to 350 Hz for flex sensors).
2. Apply linear calibration to yield joint angles ($\theta_{i}(t)$).
3. Align the temporal stream $\theta_i(t)$ with external robot pose/effector state as required.
4. Record sequences $y_{1:T}$ for each demonstration.

3. Model Construction, Calibration, and Trajectory Encoding

After collecting demonstration trajectories, probabilistic trajectory learning proceeds as follows:

• Trajectories are temporally normalized or aligned (e.g., via Dynamic Time Warping or direct time synchronization).
• For each time step $t$, the sample mean and covariance across demonstrations are computed:

$$\mu_t = \frac{1}{N} \sum_{n=1}^{N} y_t^{(n)}, \qquad \Sigma_t = \frac{1}{N-1} \sum_{n=1}^{N} (y_t^{(n)} - \mu_t)(y_t^{(n)} - \mu_t)^T$$

• The result is a time-indexed sequence of (mean, covariance) pairs, yielding a trajectory-level probabilistic model.

When implementing reproduction or teleoperation:

• At execution time, the controller may follow $\mu_t$ or probabilistically sample from $\mathcal{N}(\mu_t, \Sigma_t)$, optionally modulating control gains based on the local variance encoded in $\Sigma_t$.

4. Integration with Sensor Feedback and Teleoperation Control

Probabilistic trajectory representations can be directly inserted into bilateral teleoperation pipelines:

• The glove controller closes a feedback loop wherein finger angles $\theta_i(t)$ serve as joint references and robot tactile force feedback $f_i(t)$ is encoded into vibrotactile motor commands.
• For force feedback, the control law is

$u_i = \mathrm{clip}(K_f f_i, 0, u_{\max})$

where $K_f$ is calibrated per finger, and $u_i$ is the PWM command for vibrotactile actuation.

Online control strategies may exploit the covariance $\Sigma_t$ to adapt gains or filter stochastic perturbations, e.g., by lowering control authority during high-uncertainty motion segments.

5. Experimental Evaluation and Performance Metrics

Practical evaluation of probabilistic trajectory learning relies on metrics quantifying tracking fidelity, feedback reproducibility, and manipulation repeatability:

• Joint Tracking RMSE: Measures the deviation between human-commanded and robot-executed finger angles across several trials:

$$\mathrm{RMSE} = \sqrt{\frac{1}{5T} \sum_{i=1}^{5} \sum_{t=1}^{T} (q_i(t) - \theta_i(t))^2}$$

Reported performance: RMSE $\simeq$ 4.3° ± 1.1° over five trials (Rueckert et al., 2015).

• Force-Feedback Fidelity: Correlation coefficient $\rho$ between robot sensor force $f_i$ and glove-motor PWM $u_i$:

$\rho \text{ (feedback)} \simeq 0.92$

High linearity demonstrates effective mapping.

• Task Repeatability: Standard deviation of autonomous reproductions (σ_x, σ_y) on stacked cup placement: ≈ (5 mm, 7 mm).

6. Cost Analysis, System Comparisons, Limitations

The sensor glove system for probabilistic trajectory learning is characterized by its accessibility and easy reconfigurability:

• Total system cost: ~€250, substantially lower than commercial alternatives (e.g., CyberGlove IV ~€40,000; 5DT Data Glove ~€1,000) (Rueckert et al., 2015).
• Trade-off: Reduced sensor resolution (10-bit ADC, five DoF) and simple vibrotactile feedback modules, but full open-source hardware/software blueprint.

Identified limitations and recommendations:

• Linear resistive bend sensor calibration is straightforward but limits angular resolution near neutral posture—nonlinear models or alternative sensing (e.g., hall-effect, optical encoders) may improve fidelity.
• Current vibrotactile feedback lacks directional specificity; future integration of normal-shear actuators could enhance haptic feedback.
• IMU integration at the wrist could remove dependence on external motion capture.
• Utilizing covariance information from the probabilistic trajectory could enable variance-dependent adaptive control gains in future versions.

7. Significance in Learning from Demonstration and Future Implications

Probabilistic trajectory learning enables robust skill transfer under real-world variability and hardware constraints, particularly for bilateral teleoperation tasks where sensor gloves serve as both input and haptic feedback device. By encoding the generative process underlying human motor skills, the probabilistic approach supports the learning of complex manipulation policies, facilitates repeatable task execution, and provides paths for adaptive feedback mechanisms, laying groundwork for future advances in imitation learning and compliant robot manipulation. The approach described in (Rueckert et al., 2015) demonstrates a cost-effective, modular foundation with empirical validation for both tracking accuracy and feedback linearity, and suggests natural directions for improvement through richer sensory, higher-fidelity modeling, and adaptive control.