A Nonlinear Dynamics-Based Estimator for Functional Electrical Stimulation: Preliminary Results From Lower-Leg Extension Experiments

Abstract

Miniature inertial measurement units (IMUs) are wearable sensors that measure limb segment or joint angles during dynamic movements. However, IMUs are generally prone to drift, external magnetic interference, and measurement noise. This paper presents a new class of nonlinear state estimation technique called state-dependent coefficient (SDC) estimation to accurately predict joint angles from IMU measurements. The SDC estimation method uses limb dynamics, instead of limb kinematics, to estimate the limb state. Importantly, the nonlinear limb dynamic model is formulated into state-dependent matrices that facilitate the estimator design without performing a Jacobian linearization. The estimation method is experimentally demonstrated to predict knee joint angle measurements during functional electrical stimulation of the quadriceps muscle. The nonlinear knee musculoskeletal model was identified through a series of experiments. The SDC estimator was then compared with an extended kalman filter (EKF), which uses a Jacobian linearization and a rotation matrix method, which uses a kinematic model instead of the dynamic model. Each estimator's performance was evaluated against the true value of the joint angle, which was measured through a rotary encoder. The experimental results showed that the SDC estimator, the rotation matrix method, and EKF had root mean square errors of 2.70°, 2.86°, and 4.42°, respectively. Our preliminary experimental results show the new estimator's advantage over the EKF method but a slight advantage over the rotation matrix method. However, the information from the dynamic model allows the SDC method to use only one IMU to measure the knee angle compared with the rotation matrix method that uses two IMUs to estimate the angle.

Fig. 1

Leg extension musculoskeletal model. Five coordinate systems are displayed where subscripts: i, th, sh and g represent the IMU, thigh, shank and global frames respectively.

Fig. 2

(a) The block diagram of the SDC estimator method using n SDC parameterizations. The number of parameterizations can be increased as per a user’s choice. In the paper, 4 SDC parameterizations were used. The k subscript represents time. (b) The block diagram of the EKF estimator method. (c) The block diagram of the Rotation Matrix method.

Fig. 3

(a) The hip flexion/extension rotations used to construct the IMU to body matrix. (b) The image above shows the experimental setup on an able bodied participant.

Fig. 4

Stimulation current ramp used to calculate the saturation and threshold current amplitudes (I_t and I_s).

Fig. 5

(a) (b) The push/pull test used to determine the passive stiffness (d₁, d₃, d₄, d₅,and d₆) and mass parameters (m and l_c). (c) Isometric contractions test that determined the torque-angle (c₀, c₁, and c₂), activation time constant (T_a) and muscle activation (a_ke) parameters. (d) Pendulum test used to calculate the damping and inertial parameters (d₂ and α).

Fig. 6

The final sinusoidal input test used to determine the force-velocity parameter (c₃).

Fig. 7

The normalized stimulation input used for the leg extension test.

Fig. 8

(a) Knee joint angle estimation comparison of the left leg of person 2-trial 3. (b) Plot comparing ${\widehat{x}}_2$ of each estimator to the filtered time derivative of the encoder signal. (c) The plot of ${\widehat{x}}_3$ of each estimator over the testing period.

Fig. 9

Result of Wilcoxon signed rank test.