# US 7,080,055 B2 · Backlash compensation with filtered prediction in discrete time nonlinear systems by dynamic inversion using neural networks

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- Number: US 7,080,055 B2
- Document ID: US7080055B2
- Status: Granted
- Kind: B2
- Application number: 09/969,549
- Filed: 2001-10-02
- Published: 2004-01-22
- Granted: 2006-07-18
- Earliest claimed: 2000-10-03
- Assignees: Board of Regents, The University of Texas System
- Inventors: Javier Campos; Frank L. Lewis
- Web page: https://patentgrep.com/patent/US7080055B2

## CPC

- G05B 19/404: characterised by control arrangements for compensation, e.g. for backlash, overshoot, tool offset, tool wear, temperature, machine construction errors, load, inertia (G05B19/19, G05B19/41 take precedence)
- G05B 2219/41046: Ffw compensation using adaptive inverse backlash model
- G05B 2219/41054: Using neural network techniques
- G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G)

## Abstract

Methods and apparatuses for backlash compensation. A dynamics inversion compensation scheme is designed for control of nonlinear discrete-time systems with input backlash. The techniques of this disclosure extend the dynamic inversion technique to discrete-time systems by using a filtered prediction, and shows how to use a neural network (NN) for inverting the backlash nonlinearity in the feedforward path. The techniques provide a general procedure for using NN to determine the dynamics preinverse of an invertible discrete time dynamical system. A discrete-time tuning algorithm is given for the NN weights so that the backlash compensation scheme guarantees bounded tracking and backlash errors, and also bounded parameter estimates. A rigorous proof of stability and performance is given and a simulation example verifies performance. Unlike standard discrete-time adaptive control techniques, no certainty equivalence (CE) or linear-in-the-parameters (LIP) assumptions are needed.

## Claims

### Claim 1

A discrete-time adaptive neural network compensator for compensating backlash of a mechanical system, comprising:
a feedforward path;
a proportional derivative tracking loop in the feedforward path;
a filter in the feedforward path;
a neural network in the feedforward path and coupled to the filter, the neural network configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system; and
wherein a tracking error r(k), a backlash estimation error {tilde over (τ)}(k), and a weight estimation error {tilde over (W)}(k) of the neural network are each weighted in the same Lyapunov function.

### Claim 2 (depends on 1)

The compensator of claim 1, wherein the tracking error, the backlash estimation error, and the weight estimation error are uniformly ultimately bounded.

### Claim 3 (depends on 1)

The compensator of claim 1, wherein unknown backlash parameters are learned in real time.

### Claim 4 (depends on 1)

The compensator of claim 1, wherein the mechanical system comprises an actuator or robot.

### Claim 5

A discrete time adaptive neural network compensator for compensating backlash of a mechanical system, comprising:
a filter in a feedforward path;
a neural network in the feedforward path, the neural network configured to compensate the backlash by estimating an inverse of the backlash and applying the inverse to an input of the mechanical system; and
means for tuning the neural network in discrete time without a certainty equivalence assumption.

### Claim 6 (depends on 5)

The compensator of claim 5, wherein unknown backlash parameters are learned in real time.

### Claim 7 (depends on 5)

The compensator of claim 5, wherein the mechanical system comprises an actuator or robot.

### Claim 8

A method for compensating backlash in a mechanical system, comprising:
estimating an inverse of the backlash using a discrete-time neural network in a feedforward path;
weighting a tracking error r(k), a backlash estimation error {tilde over (τ)}(k), and a weight estimation error {tilde over (W)}(k) of the neural network in the same Lyapunov function; and
applying the inverse to an input of the mechanical system to compensate the backlash.

### Claim 9 (depends on 8)

The method of claim 8, wherein the tracking error, the backlash estimation error, and the weight estimation error are uniformly ultimately bounded.

### Claim 10 (depends on 8)

The method of claim 8, wherein unknown backlash parameters are learned in real time.

### Claim 11 (depends on 8)

The method of claim 8, wherein the mechanical system comprises an actuator or robot.

### Claim 12

A discrete-time method of adaptively compensating backlash in a mechanical system, comprising:
estimating an inverse of the backlash using a neural network in a feedforward path;
adjusting weights of the neural network using an algorithm to achieve closed loop stability without a certainty equivalence assumption; and
applying the inverse to an input of the mechanical system to compensate the backlash.

### Claim 13 (depends on 12)

The method of claim 12, wherein a tracking error, backlash estimation error, and weight estimation error are uniformly ultimately bounded.

### Claim 14 (depends on 12)

The method of claim 12, wherein unknown backlash parameters are learned in real time.

### Claim 15 (depends on 12)

The method of claim 12, wherein the mechanical system comprises an actuator or robot.