Adaptive Control of Mechanical Manipulators

John J. Craig, Ping Hsu, S. Shankar Sastry

Seminal June 1987 · The International Journal of Robotics Research

TL;DR An adaptive computed-torque controller that estimates unknown mass and friction parameters on-line, with a Lyapunov proof that all signals stay bounded and tracking errors go to zero. On a simulated two-link arm, link masses (3 kg guessed, 4 kg true) and Coulomb friction (0 guessed, 2 Nm true) converge within several seconds.

What they do

Computed torque decouples and linearizes a manipulator only if its model parameters match the real arm, yet loads and friction are often unknown and change over time. This paper adds an adaptive element: the dynamic structure is known, some or all mass and friction parameters are unknown within known (possibly very loose) bounds, and only the unknown ones are estimated on-line. The authors believe this is the first globally convergent, rigorous stability proof for such a scheme in its nonlinear setting.

How it works

Control law. Torque is the estimated mass matrix times (desired acceleration + Kv·velocity error + Kp·position error), plus estimated centrifugal, Coriolis, gravity and friction terms. With exact parameters, each joint becomes an independent second-order system.
Adaptation law. The dynamics are linear in the parameters, so servo error is driven by parameter error through known functions of the trajectory. Each joint’s error is filtered as e + αė (α chosen to make a transfer function strictly positive real), and a Lyapunov function in servo and parameter errors yields the update law. Estimates leaving their bounds are reset to the bound, which keeps the estimated mass matrix positive definite and happens only finitely often.
Convergence. Position and velocity errors go to zero; parameters converge too if the desired trajectory is persistently exciting.

Results

Simulation only: a two-link arm with viscous and Coulomb friction and random torque noise, six unknown parameters, and a sum-of-sinusoids trajectory.
All parameters start with substantial errors and are corrected over the first several seconds: link masses from 3 to 4 kg and from 2 to 4 kg, Coulomb friction from 0 to 2 Nm and 1 Nm; viscous friction behaves similarly.
Remaining servo error matches the noise amplitude divided by the closed-loop position gain.

Limitations

No real-arm experiments; the authors say adaptation speed must be verified on hardware. The update law needs joint acceleration, which manipulators usually do not sense, though its integrating action reduces the need for good acceleration measurements.

Figures

From Figure 1: structure of the controller. Servo errors pass through Kv and Kp and are added to the desired accelera... — From Figure 1: structure of the controller. Servo errors pass through Kv and Kp and are added to the desired acceleration; the estimated mass matrix and the estimated centrifugal, Coriolis, gravity and friction terms turn this into arm torques, while an adaptive element watches the servo errors and adjusts both estimates.

From Figure 2: the two-link manipulator used in simulation, with point masses at the link ends, moving in a vertical ... — From Figure 2: the two-link manipulator used in simulation, with point masses at the link ends, moving in a vertical plane under gravity.

From Figure 3: estimate of the link 1 mass over 30 seconds, adapting from an initial guess of 3 kg to the true value ... — From Figure 3: estimate of the link 1 mass over 30 seconds, adapting from an initial guess of 3 kg to the true value of 4 kg.

From Figure 7: servo error at joint 1 in radians, shrinking to small noise-driven fluctuations as the controller tune... — From Figure 7: servo error at joint 1 in radians, shrinking to small noise-driven fluctuations as the controller tunes itself.

Abstract

We present an adaptive version of the computed torque method for the control of manipulators with rigid links. The algorithm estimates parameters on-line which appear in the non-linear dynamic model of the manipulator, such as load and link mass parameters and friction parameters, and uses the latest estimates in the computed torque servo. We present what we believe is the first globally convergent, rigorous proof of the stability of such a scheme in its non-linear setting, as well as its asymptotic properties and conditions for parameter convergence. We illustrate the theory with some simulation results.

Summaries are written automatically from each paper's text. Figures and videos come from the paper and its project page; all credit goes to the authors.

Matt Strong