Inverse Dynamics Model in Robotics

• Inverse Dynamics Model (IDM) is a formulation that maps joint positions, velocities, and accelerations to actuator torques using linear regressor structures capturing inertial, Coriolis, centrifugal, and gravitational effects.
• The methodology employs least squares and closed-loop output error techniques to enable robust parameter estimation while reducing reliance on noisy position measurements.
• Experimental validation on SCARA robots demonstrates rapid convergence and high accuracy, supporting computed torque control and real-time adaptive system applications.

The inverse dynamics model (IDM) formulates the relationship between joint positions, velocities, accelerations, and the corresponding actuator torques in mechanical systems—primarily articulated robots. IDM is foundational for system identification, control synthesis, and simulation, as it encapsulates the effects of inertial, Coriolis, centrifugal, and gravitational forces in models that are often linear in parameters of interest. Accurate parameter estimation via IDM is essential for improving control performance, enabling computed torque controllers, and providing robust system models for diverse applications in both prototype and industrial robotic domains.

1. Mathematical Foundations of the Inverse Dynamics Model

The IDM is expressed as an algebraic mapping from system state variables to joint torques that are linear with respect to dynamic parameters. The general formulation presented in (Gautier et al., 2010) uses

$$T_\text{idm} = IDM_\text{st}(q, \dot{q}, \ddot{q}) X_\text{st}$$

where

• $q$ is the joint position vector,
• $\dot{q}$ and $\ddot{q}$ are the velocity and acceleration vectors,
• $IDM_\text{st}(q, \dot{q}, \ddot{q})$ is the regressor matrix (basis functions),
• $X_\text{st}$ is the vector of standard dynamic parameters (inertias, masses, centers of mass, etc).

After identifying and eliminating redundant parameters, a minimal (base) parameter vector $x$ is defined, resulting in

$T_\text{idm} = IDM(q, \dot{q}, \ddot{q})\, x$

This linear form is central to identification since it enables the construction of overdetermined linear systems that are amenable to least squares (LS) parameter estimation.

2. Parameter Identification: Classical and Closed-Loop Output Error Methods

Historically, off-line robot dynamic identification relies on sampling the IDM along trajectories that sufficiently excite the system's degrees of freedom. The standard linear identification workflow (Gautier et al., 2010) involves:

1. Tracking a reference trajectory.
2. Acquiring joint positions and estimating velocities and accelerations (typically via bandpass filtering).
3. Sampling the IDM to build a linear observation equation, relating computed and measured forces/torques.

This produces a parameter identification problem of the form $Y = IDM(q, \dot{q}, \ddot{q})\, x$, solved via LS estimation.

The paper introduces a closed-loop output error method (DIDIM) that departs from conventional practices:

• Only joint force/torque measurements are required, replacing joint position output with actuator force/torque.
• A closed-loop simulation is performed using the direct dynamic model (DDM), the reference trajectory, and the control law structure identical to the real robot.
• The output error cost function becomes

$J(x) = \|Y - y_s(x)\|^2$

where $$y_s(x) = IDM(q_{dm}(x), \dot{q}_{dm}(x), \ddot{q}_{dm}(x))\, x$$ is the simulated torque vector.

This setup leads naturally to a nonlinear LS problem, yet the crucial linearity in parameters simplifies the Jacobian in iterative solvers (e.g., Gauss–Newton), reducing computational cost:

$$\frac{\partial y_s}{\partial x} \approx IDM(q_{dm}(x), \dot{q}_{dm}(x), \ddot{q}_{dm}(x))$$

3. Measurement and Filtering Considerations

Standard identification methods demand high-fidelity joint position measurements, with velocities and accelerations estimated through numerical differentiation or bandpass filtering. This process is highly sensitive to noise, filter parameters, and sampling rate, often resulting in poor accuracy if not rigorously managed.

In contrast, the closed-loop output error approach of DIDIM (Gautier et al., 2010) eliminates the need for position-based sensor data and derivative estimation. Only direct actuator force/torque signals are required, substantially reducing sensor complexity and mitigating the impact of noisy or low-rate measurement data.

4. Experimental Validation and Comparative Performance

The DIDIM method was experimentally validated on a 2-DOF direct-drive SCARA robot. Key findings (Gautier et al., 2010):

• Rapid convergence—typically within 2–3 iterations of the nonlinear LS optimization.
• High accuracy—parameter estimates produced a torque error relative norm below 3%.
• Robustness—performance remains solid even when position measurement data is limited in sampling rate or quality, circumstances under which classical methods fail due to increased noise in velocity/acceleration estimates.

The experimental setup leveraged identical control laws and excitation trajectories for both the real and simulated robots, ensuring that the identification error reflects genuine model parameter discrepancies.

5. Computational and Practical Implications

The closed-loop output error method dramatically facilitates practical deployment:

• Computational burden is markedly reduced by leveraging the linear analytic structure of the IDM within iterative solvers; at each iteration, the LS step is tractable and quick.
• Sensor requirements are lowered—only force/torque measurements are needed, sidestepping complicated filtering and differentiation chains that introduce errors.
• Applicability is broadened to industrial settings and rapid prototyping, where robust identification under resource limitations is essential.

The methodology is suitable as a computed torque control verification tool and as a simulation engine for higher-DOF robots. While the method has been validated on a 2-DOF system, it was explicitly designed to be extensible to more complex manipulators; ongoing research is focused on 6-DOF industrial platforms.

6. Impact, Limitations, and Future Directions

The closed-loop output error identification technique (DIDIM) (Gautier et al., 2010) represents an advance in robust and efficient robot system identification. By exploiting the analytic linearity of IDM parameterization, the approach mitigates well-known pitfalls of position-based identification workflows and demonstrates resilience to typical measurement deficiencies. Limitations primarily pertain to the quality and reliability of force/torque sensors—accuracy here remains paramount.

Future research will expand DIDIM to higher dimensional systems and investigate adaptive control and real-time identification loops leveraging this simplified parameter estimation paradigm. Additionally, the seamless integration of DDM simulation within the identification process opens paths toward model-based adaptive controllers and online monitoring strategies.

---

IDM, both in classical least squares-based identification and modern closed-loop output error approaches, forms a critical foundation for the high-fidelity modeling, control synthesis, and adaptive compensation necessary in advanced robotic systems. The described innovations and experimental verifications mark a significant refinement in the practical workflow for robot dynamic identification (Gautier et al., 2010).