Complexity
Volume 2017 (2017), Article ID 5361246, 9 pages
https://doi.org/10.1155/2017/5361246

Research Article

Adaptive Neural Network Control of Serial Variable Stiffness Actuators

Zhao Guo,¹ Yongping Pan,2,3 Tairen Sun,⁴ Yubing Zhang,¹ and Xiaohui Xiao¹

¹School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China
²Department of Biomedical Engineering, National University of Singapore, Singapore
³National University of Singapore (Suzhou) Research Institute, Suzhou 215123, China
⁴Department of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China

Correspondence should be addressed to Yongping Pan; yongppan@gmail.com

Received 13 August 2017; Revised 8 September 2017; Accepted 20 September 2017; Published 8 November 2017

Academic Editor: Junpei Zhong

Copyright © 2017 Zhao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper focuses on modeling and control of a class of serial variable stiffness actuators (SVSAs) based on level mechanisms for robotic applications. A multi-input multi-output complex nonlinear dynamic model is derived to fully describe SVSAs and the relative degree of the model is determined accordingly. Due to nonlinearity, high coupling, and parametric uncertainty of SVSAs, a neural network-based adaptive control strategy based on feedback linearization is proposed to handle system uncertainties. The feasibility of the proposed approach for position and stiffness tracking of SVSAs is verified by simulation results.

1. Introduction

Variable stiffness actuators (VSAs), which can increase safety in physical human-robot interaction and meet the dynamic requirements of robots in unknown environments, have been developed for a new generation of robots [1–11]. VSAs present advantages in terms of mechanical stiffness adjusting, energy storing, and force sensing capabilities [3]. According to the design configurations, there are two typical mechanical arrangements for VSAs: one is the antagonistic configuration with a pair of actuators coupled via nonlinear springs [10], and the other is the serial configuration with two independent motors, where a principal motor drives the joint position through a compliant transmission, and a secondary motor adjusts the stiffness [11].

Antagonistic-type VSAs need two driving units to change the stiffness and position synchronously, which lead to high energy consumption and complex control design. Serial-type VSAs (SVSAs) do not have this requirement and have gained more attention. Recently, new types of SVSAs have been developed based on level arm mechanisms, where the mechanical stiffness is regulated by moving one of the level points: a pivot point, a spring located point, or a force point [12–14]. Thus, the SVSAs are not significantly influenced by the coupling between the load and the stiffness transmission mechanism. It also brings higher energy efficiency in unloaded conditions [15]. Based on the variable ratio lever principle by changing the pivot position, a novel compact rotational SVSA with an Archimedean spiral relocation mechanism (ASRM) was developed to obtain compact mechanical structure, large adjustable stiffness range, and better force transmission ability in [15]. In this innovative design, the ASRM is applied to transfer the rotation of the Archimedean spiral cam into the linear motion of the pivot, and an elastic force transmission mechanism with a spring shaft vertical to the output link is proposed to achieve large deflection angle and high energy storage ability.

VSAs are multi-input multioutput (MIMO), highly coupled, and complex nonlinear systems which include both structured and unstructured uncertainties [16]. In VSAs, the stiffness variation brings physical modifications, requiring the control system to quickly transit among different operating conditions. In addition, the coupling between stiffness and positioning mechanisms and the increased system dimensionality complicate the entire control system [16]. As a result, it is challenging to design control strategies for VSAs in robotic applications. Different control strategies have been proposed to for VSAs, where the simplest one is the PD control in [16]. However, the performance of the PD control highly depends on PD gains due to the high nonlinearity of VSAs. In [17, 18], a classical nonlinear control technique termed feedback linearization was explored for VSAs. Nevertheless, feedback linearization is very sensitive to modeling accuracy so that significant efforts in system modeling and parameter identification are required to achieve a desired performance. Based on the feedback linearization, some advanced control methods, such as gain-scheduling control [19], active damping control [20], backstepping control [21], nonlinear model predictive control [22], and output feedback control [23], have been integrated to improve stiffness and position tracking performances of VSAs. Although these control methods are proved to be effective for VSAs, they may exhibit undesired performances especially if the VSA dynamics are highly nonlinear and experience large parameter variations due to the changes of the actuator stiffness and load conditions [24].

Neural networks (NNs) have been widely applied to control complex nonlinear systems, especially complex robotic systems in recent years [25–37]. Compared with the traditional adaptive control, NN adaptive control (NNAC) has two prominent attractive features [38]. Firstly, due to the universal approximation property of NNs, the difficulty of system modeling in many practical control problems can be largely alleviated. Secondly, due to the practical persistently exciting (PE) condition of NNs, parameter convergence is easier to be obtained during control resulting in an exact online modeling and superior exponential tracking. Although NNAC is promising for complex nonlinear systems like VSAs, it is seldom applied to VSAs except [39], where antagonistic VSAs are considered in [39].

This paper focuses on modeling and control of a class of SVSAs based on level mechanisms. Firstly, a MIMO complex nonlinear dynamic model is derived to fully describe SVSAs and the relative degree of the model is determined accordingly; secondly, based on the feedback linearization, a direct NNAC strategy is proposed to govern the SVSA model; finally, high-fidelity simulations are provided to verify the effectiveness of the proposed approach. Compared with other control strategies for VSAs, the proposed NNAC can deal with high coupling and parametric uncertainties of SVSAs. In addition, the control on both position and stiffness tracking remain effective under load changing conditions.

2. Serial Variable Stiffness Actuator Modeling

As illustrated in Figure 1, a compact rotational SVSA consists of a variable stiffness mechanism (VSM), a principal motor, and a secondary motor, where the principal motor drives the output link motion through the spring transmission, and a secondary motor adjusts the stiffness of the actuator by changing the position of the pivot with an ASRM [15].

(a)

(b)

(c)

Figure 1: CAD model (a), prototype (b), and schematic model (c) of the SVSA.

By considering gravity and external loads, the dynamic model of SVSAs is represented as follows:

where is a position of the output link, with is a position associated with each motor, is a deflection angle of the elastic transmission, is an inertia of the output link, is a reflected inertia of each motor, is a reflected damping of the link, is a reflected damping of each motor, is a gravity torque, is a control input of each motor, is an external torque, is a coupling reaction torque, and is an elastic torque of the spring transmission.

The elastic torque across the transmission is given by

where is a spring stiffness, is a radius of the output link, and is a lever length ratio. The coupled flexibility torque, representing the transmission deformation reacting on the secondary motor, is given by

where is a tangent angle of the Archimedean spiral gear, is a reduction gear ratio of the secondary motor, and is a distance from the pivot point to the joint center. The stiffness of this SVSA is formulated by

The level length ratio can be represented by the position of the secondary motor as follows:

where is an initial level length ratio.

3. Neural Network Control Design

3.1. Problem Formulation

The dynamic model of SVSAs given by (1)–(5) can be represented in the standard form

where is a vector of system states, is a control input, is a system output, and

The equilibrium point of system (6) is . Let and be bounded desired outputs. Define tracking errors and .

Assumption 1. There exist uncertainties in and due to modeling difficulties and modeling errors.

Assumption 2. exist and are bounded.

The objective of this study is to design NNAC law without the exact knowledge of the SVSA model (1) such that the tracking errors and are converging.

3.2. Feedback Linearization-Based Control

Definition 3 (the standard Lie derivative notation is used for the description [40]). The relative degree from the input to the output y of system (6) at the equilibrium x₀ is if () for all and all x in a neighborhood of x₀ and () the matrix

is nonsingular at .

By simple calculation, the input relative degree of system (6) is so that can be rearranged as follows:

Therefore, there exists a diffeomorphism

that transforms system (6) into

with being a vector field.

Let and . Define

with . Then, one gets

with , , , and . A feedback linearization-based ideal control law is given by

with being a positive-definite diagonal matrix of control gains.

3.3. Adaptive Neural Network Control

As in control law (14) is not exactly known, a radial basis function- (RBF-) NN represented as follows:

is applied to approximate , where with is a matrix of NN weights, is a vector of regression functions, N is the number of neural nodes, and are commonly chosen to be Gaussian RBFs:

with and , where and are centers and widths of the Gaussian functions, respectively.

Let and . Define an optimal NN approximation error

with a constant matrix of optimal weights

According to the universal approximation property of RBF-NNs, given any small constant , there is a sufficiently large so that [41].

Assumption 4. There exists a matrix with such that holds.

Now, the actual control law is designed as follows:

and the update law of is given by

where is a positive-definite matrix of learning rates, and is a projection operator given by

Note that the choice of the gain matrix is based on a rough estimation of the inertia matrices M, , and , where the exact values of M, , and are not required in practice. It follows from the standard NNAC result [41] that system (11) driven by control law (19) with (20) achieves practical asymptotic stability in the sense that and converge to small neighborhoods of zero determined by and .

4. Numerical Results

A general specification of SVSAs in Table 1 [15] is used for simulation. The construction of the proposed NNAC follows the following steps: () to construct regression functions in (16), select three Gaussian functions to cover each universe of such that to 729; () set the filtered error parameters in (12); () set the lower bound and the control gain for the control law in (19); () set the learning rate matrix for the adaptive law in (20).

Table 1: SVSA parameter specifications.

Description
Symbol (unit)
Value

Output link inertia
(kg⋅m²)
0.0063

Motor M1 + gearbox + intermediate inertia
(kg⋅m²)
0.0134

Motor M2 + gearbox + ASRM inertia
(kg⋅m²)
0.0110

Output link damping
(N⋅ms/rad)

Motor M1 damping
(N⋅ms/rad)

Motor M2 damping
(N⋅ms/rad)

Inherent spring stiffness
Ks (N/m)
3764.5

Radius (mm)
(m)
0.05

Range of motion
(deg.)
0–360

Range of deflection angle
(deg.)
0–30

Range of stiffness
(N⋅m/rad)
1.7–150.5

A tracking task with the desired position (°) and the desired stiffness  (Nm/rad) (frequency 0.5 Hz) is chosen to show the tracking performance of the proposed controller for SVSAs, where the desired stiffness ranges between 5 and 105 Nm/rad. To make comparisons, a PD controller is selected as a baseline controller, where PD gains are optimized to minimize errors under a reasonable control u.

Tracking results by the PD control and the proposed NNAC are demonstrated in Figures 2 and 3, respectively, and comparisons of system errors are given in Figure 5, where no load is applied on the actuator in this case. It is shown that the proposed controller achieves a much higher position tracking accuracy than the PD control under a comparable control input u despite the change of the stiffness. The position tracking accuracies by the PD control and the proposed NNAC are about 0.4547° and 7.3490°, respectively. The stiffness tracking result also shows better performance of the proposed NNAC than the PD control, where the stiffness tracking accuracy reduces from 1.1200 Nm/rad to 0.2857 Nm/rad. In addition, Figure 4 shows that the norms of the NN weights and converge to some constants, which implies that an exact estimation of the system uncertainty is achieved according to the NN learning theory [38].

Figure 2: Control trajectories by the PD control without load.

Figure 3: Control trajectories by the proposed NNAC without load.

Figure 4: Evaluations of NN weights without load.

Figure 5: Comparisons of system errors without load.

To investigate the adaptability of the proposed NNAC, a 2 kg load is added to the link at t = 15 s. System errors and control inputs within 30 s are depicted in Figure 6. It is shown that, at the time 15 s, both the position error and the stiffness error increase dramatically but then reduce gradually to small values. After an adaptation about 15 s, the high position tracking accuracy is still maintained by the proposed NNAC, where the maximum position error (0.8744°) is even smaller than that before adding the load (0.9265°). The adaptability of the proposed NNAC is apparent as position error continues decreasing to small range despite the unknown load added. Note that adding loads does not affect the stiffness tracking accuracy.

Figure 6: Control trajectories by the proposed NNAC with load.

5. Conclusions

This paper has successfully applied a NNAC method to control SVSAs. Simulation results verify the ability of the proposed approach to cope with system variability, by showing remarkable control performances for both position and stiffness tracking under load changes. In future work, the practicability of the proposed NNAC would be enhanced by the composite learning [42] and the observer design [43], and experimental validity of the proposed controller will be carried out on a physical SVSA system.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61703295, 51605339, and 51675385), the Natural Science Foundation of Jiangsu Province (no. BK20150385), the Natural Science Foundation of Hubei Province (no. 2017CFB496), the China Postdoctoral Science Foundation (nos. 2016M592382 and 2017T100573), Wuhan Youth Science and Technology Dawn Plan (no. 2017050304010304), and the Fundamental Research Funds for the Central Universities (no. 2042016kf0021).

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