Calibration of an Elastic Humanoid Upper Body and Efficient Compensation for Motion Planning

TL;DR

This work introduces an elasticity-aware kinematic model for a humanoid upper body by embedding joint and transversal elasticities into an implicit torque-equilibrium framework over the DH-parameter set. It derives an exact implicit forward kinematics, solves it iteratively, and calibrates all DH parameters, elasticities, and mass distribution for Agile Justin using end-effectors tracked by an external optical system. The authors demonstrate efficient compensation by decoupling the torque-equilibrium solver from the optimization loop, achieving a substantial reduction in pose error from $21$ mm to $3.1$ mm on a broad workspace while incurring negligible runtime penalties in planning, even under significantly higher elasticities. They provide a thorough analysis of parameter contributions, calibration data requirements, and planning implications, pointing toward automatic, vision-based calibration in the future.

Abstract

High absolute accuracy is an essential prerequisite for a humanoid robot to autonomously and robustly perform manipulation tasks while avoiding obstacles. We present for the first time a kinematic model for a humanoid upper body incorporating joint and transversal elasticities. These elasticities lead to significant deformations due to the robot's own weight, and the resulting model is implicitly defined via a torque equilibrium. We successfully calibrate this model for DLR's humanoid Agile Justin, including all Denavit-Hartenberg parameters and elasticities. The calibration is formulated as a combined least-squares problem with priors and based on measurements of the end effector positions of both arms via an external tracking system. The absolute position error is massively reduced from 21mm to 3.1mm on average in the whole workspace. Using this complex and implicit kinematic model in motion planning is challenging. We show that for optimization-based path planning, integrating the iterative solution of the implicit model into the optimization loop leads to an elegant and highly efficient solution. For mildly elastic robots like Agile Justin, there is no performance impact, and even for a simulated highly flexible robot with 20 times higher elasticities, the runtime increases by only 30%.

Figures (6)

• Figure 1: Kinematic tree structure of the humanoid Agile Justin, showing its 19 degrees of freedom (red) and the mass model used for calibration (blue).
• Figure 2: The measurement setup with cameras tracking markers on the end effectors; showcasing the problem of occlusion and suitable orientation of the marker for a given robot configuration.
• Figure 3: Flowchart of the optimization loop in light gray. Dark gray shows the additional loop for the static torque equilibrium. As the outer loop converges, no additional passes through the inner torque loop are necessary.
• Figure 4: Simulated convergence towards the static torque equilibrium for robots of different stiffness. For iteration 0 the elastic effect is ignored. The bands indicate the standard deviation for 1000 different joint configurations.
• Figure 5: Histogram of the residual error at the end effectors before and after calibration, highlighting the need for non-geometric modeling of Justin. The circles on the $x$-axis mark the mean $\mu$ of the error (see also \ref{['tab:model_parameters__add_one']}).