Application of an Inertial Sensor Unit for Position Estimation and Motion Control of the Lower-Extremity Powered Exoskeleton

Introduction. The development of devices designed to facilitate motion and increase motor efficiency (of exoskeletons) is perspectives for development robotics.

Exoskeletons are increasingly used in industry [1], military [2], medicine [3], and other fields. This specifies topicality of the research on their circuit and structural design, energy consumption optimization, motion trajectory, dynamics, control.

The present paper considers an active lower-extremity powered exoskeleton of a human with a rigid structure of the load-bearing frame. It is used for the rehabilitation of the musculoskeletal patients.

The development of such exoskeletons requires solving interrelated scientific and technical problems. They include:

• formation of a skeletal scheme;
• mathematical modeling of exoskeleton motion [4–6];
• optimization of the links motion based on a mathematical model of kinematics and dynamics¹;
• reasonable construction of the motor system [7][8];
• reasonable construction of the motor system [9–11] and estimation [13][14].

All of the above tasks are connected by the problem of finding control actions for program motion. Firstly, their source may be the dynamic model of the system. Secondly, they are developed in the form of a proportional integral-differential (PID) controller used to stabilize the program motion. A combination of two methods is also possible. At that, for the formation of feedback control (in particular, in the form of a PID controller), algorithms for assessing the state of a dynamic system and a set of sensors whose indicators are used in algorithms, play an important role.

To obtain information about the position of the exoskeleton links, they are equipped with an encoder or gyroscope [9–14]. Other possible options include accelerometer, magnetometer, potentiometer, sensors of torque, force, biosignals, etc.

In [13], it is proposed to use one block of inertial sensors consisting of gyroscopes and accelerometers on each link of the exoskeleton. This solution will provide estimating the angles of rotation, angular rates and accelerations. At the same time, the measurement model turns out to be nonlinear, which complicates the structure of control and monitoring systems.

The study aimed at improving the control of the exoskeleton by using two inertial blocks attached to each link of the exoskeleton. This approach enables to estimate the rotation angles, angular rates, and accelerations of the exoskeleton. With the help of the proposed arrangement of gyroscopes and accelerometers, a linear measurement model can be obtained, which will open up ways to simplify the control system and apply the theory of optimal evaluation of linear systems [15]. This, in turn, will be the basis for improving the motion control of the exoskeleton. The criterion for improving the control quality is reducing the deviations of the angular rates of the links from the program movement.

Materials and Methods. Consider the kinematic scheme of an exoskeleton with a rigid structure of the power frame with symmetrical movement of the legs in the sagittal plane (Fig. 1).

C₂, C₃, C₄ — centers of mass of the lower legs, thighs and body, respectively; А₁, А₂, А₃ — joints connecting the links of the exoskeleton; φ₂, φ₃, φ₄ — rotation angles of the exoskeleton links; М₂, М₃, М₄ — control moments created by the actuators located in the joints А₁, А₂, А₃, respectively. Coordinate system xyz is introduced to describe the motion. x-axis is directed along the reference surface, y-axis is directed along the local vertical to the reference surface. z-axis complements the coordinate axes to the right Cartesian coordinate system.

As a basis for building an exoskeleton motion control system, we use a dynamic model from [6].

(1)

Here:

D = diag (μ₂,μ₃,μ₄) — diagonal matrix of dissipative forces characterizing linear friction in joints;

P(q) = (M_G2 cosφ₂ M_G3 cosφ₃ M_G4 cosφ₄)^T — column vector of moments arising from the action of gravity; M = (M₂ – M₃ M₃ – M₄ M₄)^T — column vector of control torques generated by actuators located in the joints of the exoskeleton links; q = (φ₂, φ₃, φ₄)^T — column vector of generalized coordinates, consisting of rotation angles of exoskeleton links; J_jk ( j, k = 2, 3, 4) — moments of inertia of exoskeleton links; μ_k (k = 2, 3, 4) — coefficients of linear friction in the joints of the exoskeleton links; M_Gk (k = 2, 3, 4) — maximum values of the moments of gravitational forces perceived by the output links of the drives.

Let the laws of change of angles and their derivatives be given for program motion:

(2)

To control the exoskeleton motion, it is proposed to create control actions in the form of a sum of control moments:

• with software management ;
• PID controller

(3)

For program control, the values of control actions M_i^P can be calculated using the formulas [7]:

(4)

Values M_i^PID are calculated according to the feedback principle (in the form of a PID controller):

(5)

or

(6)

Here, the coefficients of the PID controller K_Pi, K_it, K_Di (i = 2, 3, 4) can be:

• constant (they are found from the stability conditions);
• time functions (found from solving optimal control problems [9][13].

One of the problems of creation of control actions in the form of a PID controller is possible errors in the differentiation of measurement information. To avoid them, we use:

• encoders (for measuring the current rotation angles φ_i of the exoskeleton links);
• inertial sensor blocks of micromechanical gyroscopes (for measuring angular rates and accelerometers (for estimating angular rates and accelerations ).

The model for measuring rotation angles using encoders can be presented as:

(7)

where φ_E2. φ_E3. φ_E4 — indications of encoders located in joints А₁, А₂, А₃ (Fig. 1).

The model of angular rate measurements using micromechanical gyroscopes can be written as:

(8)

where Ω_G2, Ω_G3, Ω_G4 — indications of gyroscopes attached to the links of the exoskeleton.

To estimate angular rate and accelerations using accelerometers, a pair of two-axis accelerometers can be used, which are located at opposite ends of each of the links. They measure the apparent accelerations associated with the corresponding links in projections on x_iy_i (i = 1, 2, 3).

(9)

Here, — vectors of readings of two-axis accelerometers mounted on link A₁A₂ and located at points A₁ and A₂, respectively; — vectors of readings of two-axis accelerometers mounted on link A₂A₃ and located at points A₂ and A_3, respectively; — vectors of readings of two-axis accelerometers mounted on link A₃C and located at points A₃ and C, respectively; l₁ = |A₁A₂|, l₂ = |A₂A₃|, l₃ = |A₃C| — distances between pairs of accelerometers attached to links A₁A₂, A₂A₃, and A₃C, respectively.

We combine equations (7)–(9) and write formulas for estimating the angles of rotation, angular rates, and accelerations:

(10)

To correct estimates of angular speed modules, the equations can be used:

(11)

Thus, to estimate the rotation angles, angular rates and angular accelerations of the exoskeleton links, measurements can be processed with a moving average filter, Kalman filter algorithm, etc. [10][13][15].

In the simplest version of closing the feedback circuit when forming the control, we obtain expressions for the control moments:

(12)

The introduction of angular acceleration feedback into the control system provides the diagonalization of the inertial forces matrix to reduce the cross-influence between generalized coordinates.

We simulate the exoskeleton motions to test the operability of the proposed control law. As a program motion, consider:

• change of position from sitting to standing [7];
• stabilization of the vertical position of the exoskeleton with initial deviations of the links from the vertical.

Research Results. For numerical simulation of the system, we took the parameters of the mathematical model of the exoskeleton from [6][7]. We considered a special case by choosing the same coefficients of the PID controller for each drive: . Figure 2 shows the time dependences for the rotation angles and angular rates of the exoskeleton links. They were obtained at control moments corresponding to rising from a sitting position. As can be seen from Figure 2 a, at the end of the motion, the rotation angles of each link are equal φ₂ = φ₃ = φ₄ = 90°, which corresponds to standing.

Now let us consider the case with deviations of about 20° in the rotation angles of the links compared to the initial position set in the program motion. The simulation results of rising from a sitting position are shown in Figure 3. As in Figure 2, at the end of the motion time, the values of the rotation angles were equal to φ₂ = φ₃ = φ₄ = 90°. Consequently, the use of a combination of software control and control in the form of a PID controller accurately brought the exoskeleton to the desired position.

Thus, when modeling program motion, the operability of the proposed law of formation of control actions described in (4) and (12) was shown.

The vertical position of the exoskeleton is unstable without control; therefore, disturbing factors can cause deviations from equilibrium up to the falling of the exoskeleton.

We checked the operability of the proposed control in the task of stabilizing the vertical position of the exoskeleton. Values for the program motion were: . We considered the case when there were deviations of about 20° from the vertical position in the rotation angles of the links (Fig. 4).

Figure 4 shows that the proposed control provides stabilization of the vertical position of the exoskeleton links.

For comparison, we present the results of modeling the stabilization of the vertical position using the following values of the coefficients of the PID controller: (Fig. 5).

It can be seen from Figure 5 that when using the differential link (K_Di ≠ 0) of the regulator, it was possible to reduce emissions in the dependences of the angular rates of the exoskeleton links compared to the case K_Di = 0 (Fig. 4 b).

Thus, the possibility of controlling the lower-extremity powered exoskeleton links using the readings of encoders and blocks of inertial sensors (micromechanical gyroscopes and accelerometers) was considered. The simulation results have validated this approach.

Discussion and Conclusions. The proposed solution to the problem of controlling the motion of the exoskeleton can be used in medical applications, including for the verticalization of musculoskeletal patients.

Thus, the data of the inertial sensor blocks can be used in the task of assessing the state of the exoskeleton links — their angular rates and accelerations. At the same time, there is no need for numerical differentiation of the measuring information of the encoders. Therefore, feedback on angular accelerations can be introduced into the control system, which makes it possible to improve transients when controlling the exoskeleton motion.