Adaptive Friction Estimation

Adaptive Control based Friction Estimation for Tracking Control of Robot Manipulators

Junning Huang, Davide Tateo, Puze Liu, Jan Peters

Technische Universitat Darmstadt, German Research Center for AI (DFKI), Hessian Centre for Artificial Intelligence

Abstract

Adaptive control is often used for friction compensation in trajectory tracking tasks because it does not require torque sensors. However, it has some drawbacks: first, the most common certainty-equivalence adaptive control design is based on linearized parameterization of the friction model, therefore nonlinear effects, including the stiction and Stribeck effect, are usually omitted. Second, the adaptive control-based estimation can be biased due to non-zero steady-state error. Third, neglecting unknown model mismatch could result in non-robust estimation. This paper proposes a novel linear parameterized friction model capturing the nonlinear static friction phenomenon. Subsequently, an adaptive control-based friction estimator is proposed to reduce the bias during estimation based on backstepping. Finally, we propose an algorithm to generate excitation for robust estimation. Using a KUKA iiwa 14, we conducted trajectory tracking experiments to evaluate the estimated friction model, including random Fourier and drawing trajectories, showing the effectiveness of our methodology in different control schemes.

Demo for Drawing Path Evaluation

circle_friction_no_audio.mov

The circular motion trajectory starts at the left corner and ends at the right corner. The trajectory include four phases, three circle and one straight line. When the phase changes, the rotation of the circle inverted.

pingan_data_collect_no_audio.mov

The collection of the Chinese characters. The demonstrator draws the characters with optiTrack markers mounted on a pen holder. The four characters mean "hope you safe" in English and is in Small seal script.

pingan_friction_no_audio.mov

The robot drawing Chinese characters. The collected cartesian space trajectory from optiTrack is smoothed and filtered with a Savitzky–Golay filter then converted to joint space trajectory with inverse kinematics.

Excitation Generation

Comparison of Excitations Before and After Optimization

The excitation generated for friction estimation. The length of the trajectory is 80 seconds and it is easy to notice that the optimized trajectories cover a wider range compare with the random initial trajectories.

Effect of Standard Deviation to Number of Violation and Excitation Optimization

Left: the effect of the standard deviation (std) of the sampling distribution to the number of violation to get a feasible trajectory that satisfies joint position and velocity constraints. Right: the effect of std of the sampling distribution to the condition number of the initial trajectory and optimized excitation.

Estimation

Position Tracking Error During Estimation

Position tracking error during estimation. All tracking errors are bounded and asymptotically converge to a much smaller range compare with the beginning. The reduction of the peaks except joint 3 highlights the model's ability in compensating static friction.

Parameter Variation During Estimation

Parameter variation during estimation, from top to bottom show the coulomb friction, the difference of the breakaway force and coulomb friction, viscous friction. It's easy to see that all parameters converged.

Estimation under Different Operating Conditions

Parameter convergence under different velocity ranges. From 10% to 50% more compare with the optimized excitation trajectory.

Parameter convergence under different payloads. From 0.5kg to 1.5 kg.

The Effect of the Backstepping Design

Comparison of estimation with the backstepping design and non-backstepping design. From left to right, the absolute position tracking error for joint 1 to 7 are shown.

Evaluation with Fourier Trajectories

Fourier Trajectories for Evaluation

An example Fourier-series trajectory for evaluation, the length of the trajectory is 100 seconds.

Evaluation for Versatility of the Estimated Model

Evaluation for different controllers, highlights the model's versatility in terms of controllers. The second row shows the tracking error that filtered with the velocity threshold, the velocity threshold to filter velocity is 0.01. For each figure, from left to right, the joint position tracking error for joint 1 to 7 are shown.

Detailed Illustration of the Tracking Error

Detailed illustration of the tracking error in joint space and cartesian space for each controller. The first column shows the position tracking error, the second column shows the Log map of the left invariant cartesian space error, the third shows the norm of the Log map.

Adaptation in Non-stationary Environment

Adaptation to Joint Velocities and Temperatures

Adaptation to joint velocities that are higher than the maximum velocities of the optimal excitation, varied from 10% to 50% higher. The plot also demonstrate the adaptation to temperatures.

Adaptation to Payloads

Adaptation to different payloads, range from 0.5 kg to 1.5 kg.

Evaluation with Drawing Path

Data for Drawing Evaluation

Data for drawing evaluation. On the left figure shows the circular motion, which starts at the blue cross and ends at the green. It's composed of four phases, denote with different colors, the arrows indicate the rotation of the trajectory. On the right shows the data collection of the calligraphy drawing path with OptiTrack.

Evaluation Result of the Circular Drawing Path

Evaluation result of the circular motion, the evaluation is run with a PD controller and a PD controller with our estimated friction model. The red boxes denote the key difference of the two figures.

Evaluation Result for Calligraphy Drawing

Evaluation result of the calligraphy drawing, the evaluation is run with a PD controller and a PD controller with our estimated friction model. The red boxes denote the key difference of the two figures.

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