# A comparison of classical and learning controllers

Abstract: This paper focuses on evaluating Locally Weighted Projection Regression (LWPR) as an alternative control method to traditional model-based control schemes. LWPR is used to estimate the inverse dynamics function of a 6 degree of freedom (DOF) manipulator. The performance of the resulting controller is compared to that of the resolved acceleration and the adaptive computed torque (ACT) controller. Simulations are carried out in order to evaluate the position and orientation tracking performance of each controller while varying trajectory velocities, end effector loading and errors in the known parameters. Both the adaptive controller and LWPR controller have comparable performance in the presence of parametric uncertainty including friction. The ACT controller outperforms LWPR when the dynamic structure is accurately known and the trajectory is persistently exciting.

## Summary (2 min read)

### 1. INTRODUCTION

- The use of robotics worldwide is most prevalent in the industrial setting where the environment is highly controlled.
- Furthermore, uncertainties in the physical parameters of a system may be introduced from discrepancies between the manufacturer data and the actual system (Ayusawa et al., 2008).
- More recently, statistical regression approaches have been used to infer the optimal structure to describe the observed data, making it possible to encode nonlinearities whose structure may not be well-known.

### 2. OVERVIEW OF MODEL-BASED CONTROL

- This term globally linearizes and decouples the system, and thus a linear controller can be applied for the feedback term, uFB, which provides stability and disturbance rejection.
- Desirable performance of the computed torque approach is based on the assumption that values of the parameters in (1) match the actual parameters of the physical system.
- Otherwise, imperfect cancelation of the nonlinearities and coupling occurs.

### 3. LEARNING INVERSE DYNAMICS

- While the adaptive approach requires accurate knowledge of the structure of the dynamic model of the manipulator, the learning approach obtains a model using measured data (Nguyen-Tuong et al., 2009), allowing unknown or unmodeled nonlinearities such as friction and backlash to be accounted for.
- In order to be practical for manipulator control, learning algorithms must process continuous streams of training data to update the model and predict outputs fast enough for real-time control.
- K∑ k=1 wikŷik/ K∑ k=1 wik (8) where K is the number of linear models, ŷik is the prediction of the ikth local linear model given by (6) which is weighed by wik associated with its receptive field.
- This prediction is repeated i times for each dimension of the output vector y.
- To reduce computational effort, LWPR assumes that the data can be characterized by local low-dimensional distributions, and attempts to reduce the dimensionality of the input space X using Partial Least Squares regression (PLS).

### 4. SIMULATIONS

- In order to evaluate the performance of the LWPR learning controller, two ‘classical’ controllers in the joint space were also implemented: the resolved acceleration (RA) controller (Sciavicco and Scicliano, 2000) and the adaptive computed torque (ACT) controller (Craig et al., 1986; Ortega and Spong, 1988) given by (3).
- The LWPR controller was trained for 60s on the 0.25Hz trajectory, after which training was stopped and tracking performance was evaluated.
- Training was stopped when the observed MSE had asymptotically decreased to a low value.
- The same conditions were repeated on the ACT and RA controller.
- The third simulation adds varying amounts of error in the inertia parameters of the model while observing the resulting performance of the three controllers when tracking the ‘figure 8’ trajectory.

### 4.1 Parameter Tuning and Initialization

- The stability of the ACT controller was found to be highly sensitive to the adaptive gain parameter, γ (3).
- The initial value for the distance parameter D (7) dictates how large a receptive field is upon initialization.
- This parameter was generally tuned through a trial-and-error process which involved monitoring the MSE of the predicted values during the training phase.
- The initial performance of the LWPR controller is also highly dependent upon the data sets that are used to train the LWPR model.
- Because LWPR is a local learning approach, it must be trained in the region(s) of input space that the manipulator will be operating in.

### 4.2 Results

- The LWPR model was trained on the ‘figure 8’ trajectory at 0.25Hz, enabling it to predict the necessary torques for tracking at frequencies near 0.25Hz.
- Table 3 illustrates that the inaccurate knowledge of the dynamic parameters causes significant degradation in performance for the RA controller, due to the imperfect linearization of the system dynamics.
- For the LWPR controller, similar findings to that of simulations one and two were observed in that inertia parameter perturbations greater than 10% were sufficient to prevent LWPR from predicting accurate joint torques.
- The first test involved using the model that was learned in simulation three to attempt to track the PE trajectory.
- Unlike the adaptive controller which relies on persistence of excitation for tracking performance, the LWPR approach can be trained on an arbitrary trajectory provided that it is given sufficient time to learn.

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##### Citations

473 citations

15 citations

### Cites background from "A comparison of classical and learn..."

...While adaptive control can provide an online estimate of the dynamic parameters, is still reliant upon adequate knowledge of the structure of the dynamic model and is thus particularly susceptible to the effects of unmodeled dynamics [13]....

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...A comparison between learning approaches such as LWPR and classical control techniques [13] shows that both the adaptive controller and LWPR controller have comparable performance in the presence of parametric uncertainty....

[...]

12 citations

### Cites background or methods from "A comparison of classical and learn..."

...A technique for improving the initial learning performance by making use of full or partial knowledge of the rigid body model is applied (de la Cruz et al. (2011b))....

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...However, both dynamic parameter estimation and adaptive control methods assume a known dynamic model structure, and can be sensitive to error when the structure is not modeled accurately (de la Cruz et al. (2011a))....

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...In order to improve the generalization performance of LWPR, an algorithm for incorporating a-priori knowledge from the RBD model (1) into the LWPR algorithm is applied (de la Cruz et al. (2011b))....

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...The same motor babbling strategy as in (de la Cruz et al. (2011a)) was applied for this experiment, resulting in roughly 50,000 initial training points from motor babbling....

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...Although LWPR has the ability to learn in an online, incremental manner due to its local learning approach, performance deteriorates quickly as the system moves outside of the region of state space it has been trained in (de la Cruz et al. (2011a))....

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10 citations

### Cites background or methods from "A comparison of classical and learn..."

...In some cases it has been shown that learning can be accelerated by initializing the model with random data generated through a motor babbling process [28], [29]....

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...Random initialization We also separately initialize learning with random data generated in the motor babbling session as a benchmark, denoted ‘random’, as it has previously been shown to accelerate learning [28], [29]....

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##### References

11,343 citations

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2,966 citations

### "A comparison of classical and learn..." refers background in this paper

...Rather than learning the underlying model structure of the system, Iterative learning control (ILC) (Arimoto et al., 1984; Bristow et al., 2006) incorporates information from error signals in previous iterations to directly modify the control input for subsequent iterations....

[...]

...However, ILC is limited primarily to systems which track a specific repeating trajectory and are subject to repeating disturbances (Bristow et al., 2006), whereas the model learning approaches such as LWPR can be incrementally trained to deal with non-repeating trajectories....

[...]

2,545 citations

### "A comparison of classical and learn..." refers background in this paper

...In practice, obtaining such a model is a challenging task which involves modeling physical processes that are not well understood or difficult to model, such as friction (Armstrong-Hélouvry et al., 1994) and backlash....

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...In practice, obtaining such a model is a challenging task which involves modeling physical processes that are not well understood or difficult to model, such as friction (Armstrong-Hélouvry et al., 1994) and backlash....

[...]

2,261 citations

### "A comparison of classical and learn..." refers background in this paper

...Rather than learning the underlying model structure of the system, Iterative learning control (ILC) (Arimoto et al., 1984; Bristow et al., 2006) incorporates information from error signals in previous iterations to directly modify the control input for subsequent iterations....

[...]

...However, ILC is limited primarily to systems which track a specific repeating trajectory and are subject to repeating disturbances (Bristow et al., 2006), whereas the model learning approaches such as LWPR can be incrementally trained to deal with non-repeating trajectories....

[...]

...limited primarily to systems which track a specific repeating trajectory and are subject to repeating disturbances (Bristow et al., 2006), whereas the model learning approaches such as LWPR can be incrementally trained to deal with non-repeating trajectories....

[...]

2,033 citations

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##### Frequently Asked Questions (2)

###### Q2. What are the contributions mentioned in the paper "A comparison of classical and learning controllers" ?

This paper focuses on evaluating Locally Weighted Projection Regression ( LWPR ) as an alternative control method to traditional model-based control schemes.