Gripper-Centric Inverse Dynamics Model (GC-IDM)

• GC-IDM is a nonparametric framework that employs Gaussian process regression and expectation-maximization clustering to model and compensate for gripper-induced dynamics.
• It integrates multimodal clustering to automatically identify distinct gripper modes, achieving high accuracy in mode detection and precise impedance rendering.
• Experimental results show reduced RMS torque and tracking errors while preserving passivity, ensuring robust and safe interaction across varying tool attachments.

The Gripper-Centric Inverse Dynamics Model (GC-IDM) is a nonparametric framework for accurately modeling, compensating, and clustering the multimodal load dynamics introduced by interchangeable grippers or tools in serial manipulators equipped with joint-torque sensing. GC-IDM leverages Gaussian process regression and sampling-based expectation-maximization (EM) clustering to identify and compensate for the inertial, Coriolis, and frictional effects associated with various gripper attachments. The approach allows precise end-effector impedance rendering and maintains passivity under interaction with unknown environments, while providing automatic detection and adaptation to distinct gripper or tool modes (Haninger et al., 2019).

1. Mathematical Foundations of GC-IDM

GC-IDM begins with the rigid-body inverse dynamics equation for a manipulator with a variable gripper:

$$M(\theta)\ddot{\theta} + C(\theta, \dot{\theta})\dot{\theta} + g(\theta) + \tau_g(\theta, \dot{\theta}, \ddot{\theta}) = \tau + J_{\mathrm{int}}^T(\theta)F_{\mathrm{int}},$$

where $\theta \in \mathbb{R}^n$ denotes joint angles, $\tau \in \mathbb{R}^n$ are actuator torques, and $F_{\mathrm{int}} \in \mathbb{R}^m$ is the interaction wrench at the end-effector. The term $\tau_g(\theta, \dot{\theta}, \ddot{\theta})$ captures load-side dynamics from the gripper, including inertial, Coriolis, and frictional contributions.

To estimate $\tau_g$ from data, observations are collected in tuples $$x_t = [\theta_t^T, \dot{\theta}_t^T, \ddot{\theta}_t^T]^T$$ with the corresponding "residual" gripper torque:

$$\tau_{g,t} = \tau_t - [M(\theta_t)\ddot{\theta}_t + C(\theta_t, \dot{\theta}_t)\dot{\theta}_t + g(\theta_t)] - J_{\mathrm{int}}^T(\theta_t)F_{\mathrm{int},t}.$$

A zero-mean Gaussian process prior with a squared-exponential kernel $k(x_i,x_j;\Theta)$ is placed over the mapping $h:\mathbb{R}^{3n} \to \mathbb{R}^n$, leading to the predictive distribution:

$\theta \in \mathbb{R}^n$0

for new queries $\theta \in \mathbb{R}^n$1. The feed-forward compensation torque is set as $\theta \in \mathbb{R}^n$2.

2. Multimodal Clustering and Mode Identification

GC-IDM accommodates the fact that gripper dynamics can switch between $\theta \in \mathbb{R}^n$3 modes, reflecting different attachments or external perturbations. A latent variable $\theta \in \mathbb{R}^n$4 is introduced per sample, and conditionally $\theta \in \mathbb{R}^n$5 follows the $\theta \in \mathbb{R}^n$6-th Gaussian process:

$\theta \in \mathbb{R}^n$7

Multimodal clustering is achieved by maximizing the marginal likelihood over latent assignments $\theta \in \mathbb{R}^n$8 using an EM–SEM-Gibbs scheme. Each iteration samples $\theta \in \mathbb{R}^n$9 sequentially by leave-one-out Gibbs updates,

$\tau \in \mathbb{R}^n$0

with a simple time-correlated prior $\tau \in \mathbb{R}^n$1, $\tau \in \mathbb{R}^n$2. GP hyper-parameters for each mode are optimized by maximizing the hold-one-out log-likelihood:

$\tau \in \mathbb{R}^n$3

3. Passivity and Safety Guarantees

The overall control action is $\tau \in \mathbb{R}^n$4, with $\tau \in \mathbb{R}^n$5 a standard passive impedance controller:

$\tau \in \mathbb{R}^n$6

A storage-function analysis is applied to the closed-loop system:

$\tau \in \mathbb{R}^n$7

where $\tau \in \mathbb{R}^n$8 is gravitational potential and $\tau \in \mathbb{R}^n$9 is constructed such that $F_{\mathrm{int}} \in \mathbb{R}^m$0. The time-derivative of $F_{\mathrm{int}} \in \mathbb{R}^m$1 satisfies

$F_{\mathrm{int}} \in \mathbb{R}^m$2

demonstrating preservation of passivity despite the nonparametric feed-forward model.

4. Compensation in Impedance Control

GC-IDM compensation is performed online using the GP mean: $F_{\mathrm{int}} \in \mathbb{R}^m$3. The total torque command is

$F_{\mathrm{int}} \in \mathbb{R}^m$4

Substituting into the plant dynamics and noting $F_{\mathrm{int}} \in \mathbb{R}^m$5, the effective closed-loop dynamics are

$F_{\mathrm{int}} \in \mathbb{R}^m$6

where $F_{\mathrm{int}} \in \mathbb{R}^m$7. GC-IDM thus draws all gripper-dependent dynamics to the robot side, enabling the controller to render the desired impedance directly at the robot wrist.

5. Experimental Results

Experimental validation was conducted on a 1-DoF actuator (inertia 0.73 kg·m²) with integrated torque sensing and three interchangeable grippers of masses 0.5, 1.0, and 1.5 kg. Data were collected at 20 Hz under quasi-static PD position control.

Key results:

Test Condition	Without GC-IDM	With GC-IDM
RMS torque error, zero-impedance (lightest gripper)	0.12 Nm	0.03 Nm
RMS tracking error, pure-stiffness ($F_{\mathrm{int}} \in \mathbb{R}^m$8Nm/rad)	0.05 rad	0.015 rad
Mode clustering accuracy	–	≈98%

• Mode clustering using Gibbs-EM with $F_{\mathrm{int}} \in \mathbb{R}^m$9 modes (three grippers plus perturbation) achieved convergence in 20 iterations.
• Stability margins remained unchanged across all gripper attachments.
• Passivity assessment using mallet-induced impulses confirmed that measured power flow $\tau_g(\theta, \dot{\theta}, \ddot{\theta})$0 was always non-positive, with cumulative energy consistently absorbed by the actuator.

6. Advantages, Limitations, and Extensions

GC-IDM features several notable properties:

Advantages

• Unified, analytic-free compensation for multiple gripper/tool attachments using GP models per mode.
• Automatic identification and switching among attachments without user intervention.
• Exact rendering of desired end-effector impedance (mass and damping) despite unseen coupling effects.
• Guaranteed passivity and safety under arbitrary environment interactions through rigorous analysis.

Limitations

• Requires an explicit offline training phase for every new gripper; fully online adaptation is not realized.
• Scaling to higher-DoF systems and rich dynamic features is restricted by $\tau_g(\theta, \dot{\theta}, \ddot{\theta})$1 covariance computations.
• Compensation of static friction discontinuities is incomplete; stiction remains partially unmodeled.

Potential Extensions

• Sparse or incremental GP approaches for online identification and adaptation to novel attachments.
• Leveraging structured kernels that embed known kinematic or mass properties to accelerate learning.
• Hierarchical and continuous multimodal models for families of related grippers (e.g., with variable payload).
• Integration of vision/tactile sensing for prior-informed mode switching, such as tool recognition.

GC-IDM concretely combines multimodal GP regression, probabilistic mode identification, and passivity-preserving control synthesis to allow serial manipulators to robustly and safely interact with a range of end-effector attachments while maintaining accurate, mode-agnostic impedance control at the gripper (Haninger et al., 2019).