Joint parameter and state estimation for regularized time-discrete multibody dynamics

TL;DR

This work tackles offline joint state and parameter estimation for regularized time-discrete multibody dynamics with frictional joints by formulating a nonlinear least-squares problem that fuses inverse-dynamics and observation residuals. It solves the optimization using Levenberg-Marquardt on manifolds, leveraging a differentiable simulator with custom differentiation rules to handle complementarity and rotations. Empirical results on pendulum and Furuta pendulum hardware demonstrate fast convergence (often within seconds) and accurate identification of a high-dimensional parameter set, while identifying sensitivities to state-error weight and constraint compliance. The approach provides a practical framework to mitigate reality-gap in simulation-based control by absorbing discretization and model errors into shadow parameters, with clear directions for future uncertainty quantification and handling of non-smooth impacts.

Abstract

We develop a method for offline parameter estimation of discrete multibody dynamics with regularized and frictional kinematic constraints. This setting leads to unobserved degrees of freedom, which we handle using joint state and parameter estimation. Our method finds the states and parameters as the solution to a nonlinear least squares optimization problem based on the inverse dynamics and the observation error. The solution is found using a Levenberg-Marquardt algorithm with derivatives from automatic differentiation and custom differentiation rules for the complementary conditions that appear due to dry frictional constraints. We reduce the number of method parameters to the choice of the time-step, regularization coefficients, and a parameter that controls the relative weighting of inverse dynamics and observation errors. We evaluate the method using synthetic and real measured data, focusing on performance and sensitivity to method parameters. In particular, we optimize over a 13-dimensional parameter space, including inertial, frictional, tilt, and motor parameters, using data from a real Furuta pendulum. Results show fast convergence, in the order of seconds, and good agreement for different time-series of recorded data over multiple method parameter choices. However, very stiff constraints may cause difficulties in solving the optimization problem. We conclude that our method can be very fast and has method parameters that are robust and easy to set in the tested scenarios.

Figures (12)

• Figure 1: Abstract representation of the impulse residuals $\Delta \bm{p}_k$ and observation residuals $\Delta \bm{y}_k$. The white circles with black outlines represent the state optimization variables. These are used to predict the next states (blue diamonds) according to the discrete dynamics $\bm{\phi}$. However, the lines connecting them are dashed to indicate that this computation never takes place. Instead, the residuals $\Delta \bm{p}_k$ are computed directly using inverse dynamics. The state optimization variables also predict the observations (red diamonds). The observation residuals are given as the difference between the predicted observations and the measured observations (gray squares).
• Figure 2: Images of the pendulum and the Furuta pendulum.
• Figure 3: A render of the Furuta pendulum with annotated joint angles and lengths. The two rigid bodies are drawn separately to the right, and their respective principal components of inertia are indicated.
• Figure 4: The two upper plots show three of the estimated parameters, tuned to the swing-up scenario versus different method parameters: State error weight in the left plot and compliance in the right plot. The lower plots show the mean squared error (MSE), plotted for observation error and state error versus the $\kappa$ and $\epsilon$.
• Figure 5: The left column of plots shows simulated observation trajectories using models tuned with different $\kappa$. From top to bottom, we have measured joint angles $\theta_1$, $\theta_2$, and the control signal $u$ versus time. The right column of plots shows the parameter values for two of the tuned parameters versus iteration for the same set of $\kappa$. The bottom right plot shows how the cost decreases under optimization.