J. Cent. South Univ. (2021) 28: 140-152

DOI: https://doi.org/10.1007/s11771-021-4592-2

Nonlinear observer-based optimal control of an active transfemoral prosthesis

Anna BAVARSAD¹, Ahmad FAKHARIAN¹, Mohammad Bagher MENHAJ²

1. Faculty of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch,

Islamic Azad University, Qazvin 34185-1416, Iran;

2. Department of Electrical Engineering, Amirkabir University of Technology, Tehran 15875-4413, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: This paper designs a joint controller/observer framework using a state dependent Riccati equation (SDRE) approach for an active transfemoral prosthesis system. An integral state control technique is utilized to design a tracking controller for a robot/prosthesis system. This framework promises a systematic flexible design using which multiple design specifications such as robustness, state estimation, and control optimality are achieved without the need for model linearization. Performance of the proposed approach is demonstrated through simulation studies, which show improvements versus a robust adaptive impedance controller and an extended Kalman filter-based state estimation method. Numerical results confirm the benefits of our method over the above-mentioned approaches with regard to control optimality and state estimation.

Key words: state dependent Riccati equation; observer; integral state control; tracking; active transfemoral prosthesis

Cite this article as: Anna BAVARSAD, Ahmad FAKHARIAN, Mohammad Bagher MENHAJ. Nonlinear observer- based optimal control of an active transfemoral prosthesis [J]. Journal of Central South University, 2021, 28(1): 140-152. DOI: https://doi.org/10.1007/s11771-021-4592-2.

1 Introduction

Many people around the world have amputations in the lower limbs for a variety of reasons, and unfortunately their numbers are increasing. Lower limb amputations include below knee (transtibial), above knee (transfemoral), foot amputations, through a joint (hip and knee disarticulations). The ability to walk normally for the handicapped could be restored using technology and advanced prosthetic legs. Passive (without any electronic control), active (with motors) and semi-active (without motors) are the three types of prosthetic legs.

A more natural gait is offered by active prostheses compared to other types of prostheses for people with amputations, and the user needs less power and energy [1]. The amount of energy required to walk is one of the biggest challenges for people with lower limb amputations. This point shows that better and more accurate design of robotic legs will improve their quality of life. As a result, the motivation of this study is to reduce energy consumption using nonlinear optimal control, estimate state variables to improve controller performance, and eliminate large loading sensors and reduce weight of the prosthesis using a nonlinear observer, as well as improve the tracking of reference trajectories. POPOVIC et al [2] presented the integration of a second powered degree of freedom (DOF) into the knee joint mechanism to control the internal-external rotation of the shank-foot complex. The design and control of a transfemoral prosthesis is proposed using passive impedance functions [3]. A robust tracking control of a prosthesis is designed [4]. ZHAO et al [5] suggested a novel optimization-based optimal controller for AMPRO. Other controllers such as robust adaptive impedance, variable impedance (VI), and proportional–derivative (PD) have also been proposed for active prosthesis [6,7]. Recently, SDRE control and robust SDRE control have been designed for this system, assuming that all states are available and in presence of ±30% uncertainty in the parameters [8, 9].

It is usually much more difficult to design a controller for nonlinear systems than linear systems. Approximating a control system with a linear model may sometimes be appropriate, but if the range is broad, a linear controller may be unstable. Taking into account the system’s non-linearity, their effects on system performance and design uncertainties, it is also possible to understand the significance of nonlinear controllers. Cost and performance are other reasons to use nonlinear control techniques. Linear control may need precise actuators and sensors to create linear behavior in the work area; whilst, nonlinear control enables the usage of cheaper equipment with nonlinear properties [10]. Hence based on the advantages of nonlinear controllers, this article uses the SDRE-based nonlinear optimal control system, which does not involve linearization operations and the Jacobian calculations, and due to its benefits such as flexibility in design, robustness, and asymptotic stability, a systematic process with ability to combine with other control methods, and to trade- off between tracking and the amount of control effort using the weighing matrices, has attracted the attention of many scholars.

CIMEN [11] presented a comprehensive review, together with description of the SDRE technique, containing its structure, theorems of optimal stability and state-dependent parameterization. CLOUTIER [12] has reviewed several SDRE design methods such as SDRE nonlinear H_∞, SDRE nonlinear H₂, and SDRE nonlinear filtering. In the algebraic and differential mode, the SDRE technique has been designed and experimented on various robots such as 6R fixed robot (rigid joint) and three-link PUMA manipulator (flexible joint), and in some cases the feed-forward gain has been used [13, 14].

Since the SDRE controller is a state feedback, it is necessary to measure and apply all states at any time in the control structure. Therefore, for a system where not all states are measured, an observer must be designed and employed to estimate the states. One of the new methods of estimating the state of nonlinear systems is the filtering methods based on SDRE. This observer is based on the SDRE control dual feature, which has a structure similar to the steady-state Kalman filter for linear systems, except that its gain is calculated using the state-dependent Riccati equation [15]. CIMEN et al [16] studied the performance of the SDRE filter on a nonlinear system compared to the EKF. The results of which indicate the superiority of SDRE. The SDRE controller and estimator designed and experimented for 6R robot [17]. NEKOO [18] presented a complete review and tutorial on the SDRE-based controller and observer design near the end of 2017. He mentioned the application of this method for various systems. Furthermore, the combination of the SDRE and other control methods such as sliding mode, integral sliding mode, and model reference adaptive was investigated.

To estimate states in the robot/prosthesis combined system, FAKOORIAN et al [19] utilized an extended Kalman filter (EKF) for this system with four-DOF that the results showed if the initial estimation errors are too high, filter performance may be poor. Following their article, they redesigned an unscented Kalman filter (UKF) to evaluate ground reaction force (GRF) and states of this system [20]. MOOSAVI et al [21] proposed a robust derivative free Kalman filtering (DKF) for state estimation for the prosthetic leg. AZIMI et al [22] used two observers, sliding mode and adaptive, to estimate the unknown GRF for this system.

In this study, a controller and observer according to the SDRE approach is used for the active transfemoral robot/prosthesis system. Since medical applications are related to movement and optimal control, reduction of energy consumption has always been a concern. Therefore, in this paper, first, the optimal energy discussion and then the estimation of state variables to better control the system and finally the desirable tracking of the system’s positions are discussed.

The article is structured as follows. Section 2 describes the robot/prosthesis system; Section 3 discusses the SDRE controller and its formulation; in Section 4, the SDRE observer and its formulation are presented, Section 5 discusses the proposed technique on the prosthetic leg model; in Section 6, simulation results are presented; the final section is discussed conclusions.

2 Robot/prosthesis system

In this section, the robot/prosthesis system with three-rigid links and three-DOF is examined which is affected by the GRF. As can be seen in Figure 1, this system has a prismatic-revolute- revolute (PRR) structure, of which the first DOF indicates the vertical displacement of the human hip, rotational axis represents the angle of thigh (robot), and the last rotational axis is for the angle of knee (prosthesis) [1].

Figure 1 Three-DOF robot/prosthesis system with rigid ankle

A dynamic model for the robot/prosthesis system is given as:

(1)

whereand are the reversible inertial matrix, the Coriolis and Centripetal matrix, the gravity vector, and the nonlinear dumping vector, respectively (refer to Appendix); alsois the displacement vector of the joint (p₁ is hip vertical displacement; p₂ is thigh angle; p₃ is prosthetic knee angle); includes the active control force at the hip, and the active control torques at the thigh and knee; N(p) is the combined effect of the horizontal component F_x and vertical component F_z of the GRF on each joint. Assuming the prosthesis test robot walks along the horizontal axis x, a treadmill is considered the robot’s walking surface. As a result, by modeling the treadmill belt such as a mechanical stiffness, the treadmill’s reaction forces will be functions of the belt deflection [1].

The concept of N(p) of the GRF is described as:

(2)

(3)

(4)

(5)

where l₂ is the length of the thigh and l₃ is the length of the shank; L_z is the vertical position of bottom of the leg in the world frame; s_z, k_b and β are the vertical distance between the base of the frame and the belt, the belt stiffness, and the friction coefficient of the belt, respectively (as shown in Figure 1) [1]. The nominal values of model parameters and the treadmill parameters are mentioned in Ref. [9].

Equation (6) gives states, desired trajectories, and controls as:

(6)

In the following, the design of the tracking controller and observer is presented, which estimates all state variables well in addition to minimizing the energy consumption, and also p₁, p₂ and p₃ track the desired trajectories carefully.

3 SDRE controller

The SDRE nonlinear optimal controller is one of the most important controllers based on the state-dependent coefficients (SDC) parametrization method. The controller’s aim is to obtain a control input that minimizes the respective cost function by applying the control system while stabilizing the closed loop system and meeting the constraints specified for it; while the system state variables have good tracking output with the least control effort. The benefits of this technique include the lack of uniqueness of the nonlinear system’s SDC parametrization representation, the conservation of all features of the nonlinear system, the use of this method in controller and observer development, and the possibility of a reasonable compromise between control and tracking performance.

Consider the time invariant affine nonlinear system with the following state space representation:

(7)

where and are the state variables and system inputs.

Now considering and the nonlinear system (7) is represented as the state-dependent parameterized system (8):

(8)

The SDC parameterization technique is an operation by preserving the previous structure, which converts the nonlinear system into a pseudo-linear matrix system. In formula (8), and that are the SDC parameterization matrices. If the system has more than one state parameter, it is possible to perform SDC parameterization in infinitely different forms. The performance index J should be minimized in order to design the optimal control as:

(9)

where I is the weighting matrix for states (I>0, n×n), , and E is weighting matrix for control inputs (E>0, m×m). Both are symmetric.

The SDRE control law is calculated as:

(10)

in which is a square matrix (n×n), non- unique, and symmetric, and its value is obtained by solving the following state-dependent algebraic Riccati equation:

(11)

The “care” command in MATLAB software easily solves equation.

Assumption: The pairs of and are pointwise stabilizable (controllable), and pointwise detectable (observable) for all x, respectively [11].

If the rank of matrix of Eqs. (12) and (13) is full, the system’s controllability and observability are guaranteed, respectively [11].

(12)

(13)

4 SDRE observer

In most design methods, it is assumed that all state variables are available. However, in practice, it is necessary to have a sensor built-in to measure each of the unknown and unmeasured variables, which is often not feasible, requires high cost and has low accuracy. On the other hand, they make prosthetic legs heavier. To solve these problems, if the system is observable, state variables can be estimated based on the system outputs.

The nonlinear system with noise is given by:

(14)

where , and are the state vector, the input vector, and the output vector of the system, respectively. The vectors and are the uncorrelated zero- mean white Gaussian noise vectors with identity covariance.

By placing Eq. (10) in Eq. (14), it is resulted:

(15)

where A_cl(x(t)) is the stable-closed-loop matrix of the system:

(16)

Conventional SDRE estimator structure is given by Eq. (17):

(17)

whereis the state estimation vector and is the observer’s gain obtained by:

(18)

where is positive definite matrix calculated by:

(19)

where is the weighting matrix for estimation of system’s states (S≥0, r×r) and is the weighting matrix for the observer inputs (V>0, p×p), which are both symmetric.

To obtain the dual equation of SDRE filter equations, a change in the variable of Eq. (20) is considered:

(20)

where is the state vector of the system, is the output vector of the dual system, and is the input vector.

For infinite time mode, the dual system performance index is considered as follows [17, 18]:

(21)

5 Proposed method on prosthetic leg model

As noted earlier, it is shown that for multivariate structures there are countless SDCs presentations. This benefit of the SDRE approach offers design versatility that can be used to enhance system performance or balance between optimization, reliability, rejection of disruption, and robustness. It is not easy to factorize and determine SDC matrices for complex robotic systems with a high DOF. In this study, the SDC matrices are determined as the suggested structure [13, 14].

(22)

(23)

5.1 Combination of SDRE and integral state control to design a tracking controller

In this subsection, integral state control is recommended to improve tracking control. According to this technique, the integral of tracking error of the three positions is added to the system (8) such as the new state variables. Consequently, the augmented state vector x_a is obtained as follows:

(24)

In this case, the matrices A_a, B_a and C_a are set as

(25)

where matrices A and B are equal to Eqs. (22) and (23). C=I_3×3 is the system output matrix. Therefore, the state-space and output equations of the system change to Eq. (26):

(26)

As a result, the SDRE controller can be designed for this system, if and only if, for all x the rank of matrix L is complete.

(27)

Accordingly, by utilizing the SDRE method to the augmented system (26) and using Eqs. (10) and (11), the control law is calculated according to Eqs. (28) and (29).

(28)

(29)

Finally, this technique leads to asymptotic stability of the closed loop system and eliminates fixed disturbance. More details can be found in Ref. [9].

5.2 Utilizing of SDRE observer

To utilize the SDRE observer, first all the state-dependent parameters must be obtained as a function of the estimated variables which can be used in the observer equations. In the following steps, by calculating the observer gain (18), the system state space is estimated according to Eq. (30).

⁽³⁰⁾

Figure 2 indicates the block diagram of the SDRE controller/observer structure on the robot/ prosthesis system.

Figure 2 SDRE controller/observer structure on robot/ prosthesis system

The following two performance indicators are defined to evaluate the controller and observer performance in terms of tracking, estimating, and reducing energy:

(31)

where RMSE_i and RMSU_i are the root mean square tracking error for each state and the mean square value of each control signal, respectively. T is the time period. P_r, x and u are according to Eq. (6). The normalized cost function is calculated as

(32)

Finally, the values of the total desired trajectory tracking cost, total control cost, and their total cost are obtained by:

(33)

6 Simulation results

In this part, simulation results of the proposed controller and observer are given on the robot/ prosthesis system.

Since the DC motors have speed and torque limitations, as a result, some limitations are considered based on the minimum and maximum allowable values of the control inputs (saturation bounds) in the simulations, which should not be exceeded (actuator limit), as follows:

(34)

where u_i,stall is stall torque; α_i,nl is no-load speed and α_i(t) is actual speed of i-th motor [11, 14]. Consequently:

(35)

Based on this, the saturation bounds for hip displacement force is [-900, 900] N, for thigh torque is [-600, 600] N·m, and for knee torque is [-400, 400] N·m. Accordingly, the simulation results are obtained by applying these constraints to the amplitude of the control signals.

6.1 Tracking controller and observer

In the tracking section for desirable trajectories, walking data obtained by the Motion Studied Laboratory of the Cleveland Department of Veterans Affairs Medical Center, Iran have been used [1, 19].

In Table 1, the initial condition values for the controller and observer, the measurement noise covariance matrix (MN), and the process noise covariance matrix (PN) (Eq. (36)) are specified according to Refs. [1, 19].

(36)

The weighting matrices for the controller (E, I) and the observer (S, V) are set out in Table 2.

It should be noted that in this article, the main purpose is to track positions, but their velocities are bounded as figures plot.

Table 1 Values of initial conditions of controller and observer, and measurement and process noise covariance matrices

Table 2 Controller and observer parameters

Figure 3 indicates the tracking and estimation performance compared to the closed loop system states with desired trajectories. As shown, tracking of positions p₁, p₂ and p₃ is well done, and their velocities track the reference data and remain bounded. In terms of performance of the SDRE observer, it is also concluded that even with the initial non-zero error for the positions and their velocities, the estimated states converge rapidly to the correct states and the estimation performance is satisfactory. In Table 3, the numerical indices of this section are presented.

As shown in Figure 4, due to the difference between the initial conditions of the states and the maximum value of the difference occurred at the beginning of the work, the observer quickly brings the estimated states to actual values and estimation error tends to zero, which indicates the proper performance of the SDRE observer.

Figure 5 depicts the system control signals for the values of the nominal parameters with and without the observer considering the saturation bound. As it is evident, at the beginning of the movement due to the difference with the starting points of the desired trajectories and because of the error caused by the hip displacement force, the thigh and knee torques reached their saturation bound with and without the observer. However, after about 0.2 s, the amplitude of these control signals decreases and falls within the permissible range. Figure 5 also shows that the control magnitudes with and without observer are similar. This illustrates that the tracking performance with and without observer is still at acceptable levels.

Table 3 lists a summary of the numerical indices RMSE_i and RMSU_i using Eq. (31); Cost_E, Cost_U, and the total cost based on Eqs. (32) and (33) are calculated. This table also shows the results of Refs. [1, 19] for comparison. As it can be seen, the proposed method has a better performance in the energy-consumption so that it reduced Cost_U by 67.447% compared to Ref. [1]. It also has a relatively desirable performance in tracking. Analyzing the tracking performance shows that Cost_E of the proposed method is 43.673% lower than the result in Ref. [1]. From the total cost point of view, according to the formula given in Ref. [1], the proposed method in this paper reduced the total cost by 52.25% compared to Ref. [1]. As illustrated by the numerical indices of this table, this method also performs very well in the state estimation discussion compared to the results of Ref. [19] in this test. It should be noted that the total costs for each of the six states (positions and velocities) are measured in this study in order to equate this paper with Ref. [1], and although the velocity tracking was not done correctly, the total cost of work is lower than Ref. [1].

Figure 3 SDRE tracking and estimation performance:

Table 3 Numerical indices of RMSE_i, RMSU_i , Cost_E, Cost_U, and Cost compared to other methods

Figure 4 Estimation errors:

Figure 5 Control signals with and without observer:

6.2 Convergence

In this section, according to Ref. [19] convergence experiments have been performed to investigate the performance of the SDRE observer compared to the EKF in that paper. For this purpose, the initial values and x(0), and the measurement and process noise covariance matrices (MN and PN) in two tests are selected as the values shown in Table 4. The results are also mentioned in this table.

Table 4 Initial values, covariance matrices noise terms, and error behavior of observer

It is noteworthy that, for both tests, the thigh angle and the thigh angular velocity are depicted, although other states have similar and desirable results.

6.2.1 Test 1

Figure 6 shows the performance of the SDRE observer in presence of the small initial error and small noise. As can be seen, the estimation error is bounded.

6.2.2 Test 2

In the last test, the error behavior of the observer in the discussion of system state estimation is examined in presence of large initial error and small noise terms. As shown in Figure 7, the performance is satisfactory and desirable, and contrary to the simulation results of Ref. [19], in this test, the estimation error does not diverge and remains bounded. This indicates the proper performance of the SDRE observer.

Remark: Implementation of SDRE controller and observer could be done through “Arduino” which is explained completely in Ref. [23].

7 Conclusions

The SDRE technique was used as a controller and observer for the prosthetic leg for transfemoral amputees. First of all, the main goal of the paper was to estimate all state variables using an SDRE-based observer, secondly to reduce the energy consumption, and finally to improve tracking of the positions by combining SDRE with integral state control technique. To do this, the state-space equation of nonlinear system was converted to the SDC matrices form. After determining the weighting matrices and solving the algebraic Riccati equations, the control law was obtained and utilized for the robot/prosthesis system. Simulations were demonstrated by considering the saturation bound of control inputs according to limitation of actuators. The results show that the proposed method significantly reduces the energy consumption compared to a robust adaptive impedance controller and has an acceptable tracking performance. Numerical indicators have shown that the total cost of the proposed approach is lower than the robust adaptive impedance control. This paper also compared the performance and convergence of the SDRE estimator and the EKF in line with the previously published results, which showed the superiority of the proposed SDRE observer even in spite of large initial value errors.

Figure 6 Observer performance and estimation error in presence of small initial error and small noise:

Figure 7 Observer performance and estimation error in presence of large initial error and small noise:

Since there was no access to the actual system, an attempt was made to use the original data provided by the Cleveland State University to bring the results closer to reality. In future work, the tracking controller will be designed for both positions and velocities. To increase the robustness of the closed loop system against the uncertainties of the model, a robust portion will also be added to the proposed controller structure.

Appendix

; (A1)

; (A2)

; (A3)

(A4)

. (A5)

Contributors

The initial idea generation was by Ahmad FAKHARIAN. The design of the control system and estimator, simulations, data analysis, and the initial draft of manuscript was done by Anna BAVARSAD. Review of designs and simulation results and editing of the article by Ahmad FAKHARIAN and Mohammad Bagher MENHAJ.

Conflict of interest

Anna BAVARSAD, Ahmad FAKHARIAN, and Mohammad Bagher MENHAJ declare that they have no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

基于非线性观察者的主动股骨假体的最优控制

摘要：本文使用状态依赖的Riccati方程(SDRE)方法为主动经股假体系统设计联合控制器/观察器框架。利用积分状态控制技术来设计机器人/假体系统的跟踪控制器。该框架保证了系统灵活设计，通过该设计，可以实现多种设计参数优化，例如鲁棒性、状态估计和控制最佳性，而无需模型线性化。仿真研究证明了该方法的性能，较鲁棒的自适应阻抗控制器和基于扩展卡尔曼滤波器的状态估计方法有所改进。数值结果证实了本方法在控制最优性和状态估计方面优于上述方法。

关键词：状态依赖的Riccati方程；观察者；积分状态控制；追踪；主动式股骨假体

Received date: 2020-01-19; Accepted date: 2020-07-22

Corresponding author: Ahmad FAKHARIAN, PhD, Associate Professor; Tel: +98-28-33665275; E-mail: ahmad.fakharian@qiau.ac.ir; ORCID: https://orcid.org/0000-0002-6530-2689