Adaptive feedforward control of a collaborative industrial robot manipulator using a novel extension of the Generalized Maxwell-Slip friction model

Highlights

• GMS model is extended (E-GMS) with dependency on joint torque and angular position.

• E-GMS based adaptive feedforward control compensate unmodeled wear and temperature.

• Implementation on an industrial robot yield superior torque prediction and tracking.

Abstract

Collaborative industrial robots often use strain-wave transmissions which display a highly nonlinear behavior. In particular, the friction torque depend on the load torque and the hysteresis characteristics were recently found to depend on the joint angular position.

This paper presents a novel extension of the Generalized Maxwell-Slip friction model to describe said phenomena in a combined framework. The method overcomes the discontinuity around zero velocity of existing models. Experiments on the Universal Robots UR5e manipulator show superior performance in terms of torque prediction accuracy and tracking performance of the proposed method.

An adaptive feedforward friction compensator is proposed based on the extended Generalized Maxwell-Slip friction model to compensate the time-variations of the Coulomb and viscous friction due to, respectively, wear and mispredictions of the lubricant temperature. The adaptive estimator relies on the sensing hardware readily available in the joints of the Universal Robots manipulators, i.e. two absolute rotary encoders; one at each side of the transmission, current sensing for the electric actuator, and a temperature sensor. Results show a considerable reduction of the torque prediction error and tracking error.

Introduction

For industrial robot manipulators, and in particular collaborative robots, the ability to accurately predict the actuator torques required to realize the desired task is highly important. An increased accuracy of torque predictions generally lead to improved robot performance, specifically enhanced accuracy in motion and force control tasks [1], [2], smoother lead-through programming¹ experience [3], [4], and increased performance of robot safety systems [5], [6], [7]. Additional benefits include lower energy consumption and the possibility to use hardware of reduced cost, e.g. less accurate sensors.

Collaborative robots most often utilize strain-wave type transmissions such as the Harmonic Drive™ [8] due to their desirable characteristics of high torque capacity and low weight. However, their inherent complex nonlinear friction and hysteresis characteristics complicates the accurate mathematical modeling and thus leads to decreased robot performance if not modeled and properly compensated.

In addition to the challenge of accurate joint dynamics modeling exists another challenge in which the changes in ambient temperature and the wear and tear of the robot joints cause time-variation of the friction characteristics and thus introduce errors and uncertainties to the mathematical models, which leads to decreased robot performance if uncompensated. The temperature affects the viscous friction [9] while the wear and tear affects the Coulomb friction [10].

The above-mentioned challenges of accurate joint dynamics modeling and time-varying friction characteristics motivates the use of: 1) Dynamic friction compensation to compensate the complex nonlinear dynamics of the strain-wave transmissions, and 2) adaptive control to continuously learn from the inaccuracies of the mathematical models and adapt the models to better represent the real robotic system. Such compensation is typically introduced in the feedforward part of the control structure [11], i.e. eliminating the frictional effects by adaptive feedforward dynamic friction compensation.

Friction is by nature a complex fluid dynamic phenomenon. To accurately describe the frictional behavior at near-zero velocities, dynamic friction models are required. While the LuGre model [12] have attained much interest in the general robotics community, the Generalized Maxwell-Slip (GMS) model [13], [14] have proven especially useful for modeling the friction and hysteresis characteristics of strain-wave transmissions [15], [16] due to its ability to describe the rate-independent hysteresis phenomenon. Among others, the GMS model has proven successful for describing the dynamic friction characteristics of the Universal Robots manipulators, see [17], [18].

Several studies have been conducted to characterize the dynamics of strain-wave transmissions. The steady-state friction characteristics of a strain-wave transmission is known to depend on the angular velocity, temperature, and load torque. However, existing dynamic friction models with load torque dependency does not solve the discontinuity around zero-velocity. Also, in Dong et al. [19] the effects of wave generator radial offset was investigated theoretically and the consequent effects were uneven distributions of the backlash and flexspline preload over the angular position. In [18] the magnitude of the backlash was indeed observed to depend on the angular position of the Universal Robots UR5e manipulator.

Some works presented adaptive friction compensation based on the GMS friction model. Nilkhamhang and Sano [20], [21] developed a switching adaptive controller based on the GMS model with linearized Stribeck friction function. Stability was ensured by parameter projection and numerical simulations demonstrated parameter convergence. However the choice of switching function is not suited for implementation in real systems as noted by Grami and Bigras [22]. Grami and Bigras identified the GMS friction model using the robust adaptive observer developed by [23]. The observer was verified by numerical simulation of a single-operator GMS model. In [24] their method was further improved by filtering the regressor to respect the requirement of Lipschitz continuity. However, the issue of the ideal switch was not solved. In [25], [26] a recursive least squares estimator with exponential forgetting was proposed to adapt the DNLRX² model – a GMS friction model combined with FIR filters for the state vector and input position. Experimental results on an XY positioning stage showed great performance for the adaptive DNLRX model compared to a standard PID controller. To further enhance the tracking performance of GMS based feedforward control systems, Jamaludin et al. [27] designed an inverse model-based disturbance observer. Such strategy is indeed effective for increasing the tracking control performance. However, since the robot does not learn from the time-varying friction characteristics and on-line update the friction model, the disturbance observer strategy does not improve the robot performance in terms of safety. Other observers relevant for state estimation includes high-gain observers [28], sliding-mode observers

The aforementioned studies does not solve all challenges related to accurate actuator torque prediction for collaborative industrial robot manipulators. The immediate negative consequences are related to the robot safety, motion and force control performance, and lead-through programming experience.

In this paper, we present an extension to the GMS friction model to handle in a combined framework the load torque dependent Coulomb friction and the dependency of backlash on the joint angular position. We prove that the steady-state response of the extended GMS model is equivalent to that of the original GMS model. The nonlinear viscous friction is considered to depend on angular velocity and temperature. Additionally, a new gradient-based adaptive control strategy is proposed based on the extended GMS friction model to address the time variation of the friction characteristics. The extended GMS model and adaptive feedforward compensator are validated on the Universal Robots UR5e collaborative robot manipulator.

The organization of this paper is as follows: Section 2 presents the adaptive feedforward dynamic friction compensation strategy. Next, the mathematical model of the robot manipulator is presented in Section 3, and in Section 4 the mathematical model of the robot joint dynamics is detailed. Section 5 presents the Extended GMS friction model which is allowed to depend on the joint torque and backlash. The friction model is validated on the Universal Robots UR5e robot manipulator. In Section 6 the adaptive estimator is presented, and its effectiveness is demonstrated on the UR5e robot. Section 7 concludes on the work and presents our ideas for future research.

Let $\mathbb{R}$ be the set of real numbers, $\mathbb{Z}$ the set of integers, and $\mathbb{N}$ the set of non-negative integers, denote then ${\mathbb{R}}_{+}=\{x\in \mathbb{R}:x\ge 0\}$ and ${\mathbb{N}}_{+}=\{x\in \mathbb{N}:x>0\}$; | · | denotes the absolute value and || · || is the 2-norm; $\mathbf{x}\in {\mathbb{R}}^n$ is a vector of n real numbers, x_i is the ith entry of $\mathbf{x},$ ${\mathbf{x}}^T$ its transpose, $\overline{\mathbf{x}}$ the mean value the elements of $\mathbf{x},$ let then $\widehat{\mathbf{x}}$ be an estimate of $\mathbf{x}$ and define the error vector $\tilde{\mathbf{x}}\triangleq \mathbf{x}-\widehat{\mathbf{x}}$; given a square real matrix $\mathbf{A}\in {\mathbb{R}}^{n\times n}$ let $\mathbf{A}\succ 0$ indicate that $\mathbf{A}$ is positive definite, i.e. ${\mathbf{x}}^T\mathbf{A}\mathbf{x}>0$ for any non-zero column vector $\mathbf{x}$ of n real numbers; let $\mathrm{diag}:{\mathbb{R}}^n\to {\mathbb{R}}^{n\times n}$ map a vector of n elements to a diagonal matrix with the ith element of the vector on its ith diagonal entry and zero everywhere else.