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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, we deal with Markov Jump Linear Systems-based filtering applied to robotic rehabilitation. The angular positions of an impedance-controlled exoskeleton, designed to help stroke and spinal cord injured patients during walking rehabilitation, are estimated. Standard position estimate approaches adopt Kalman filters (KF) to improve the performance of inertial measurement units (IMUs) based on individual link configurations. Consequently, for a multi-body system, like a lower limb exoskeleton, the inertial measurements of one link (e.g., the shank) are not taken into account in other link position estimation (e.g., the foot). In this paper, we propose a collective modeling of all inertial sensors attached to the exoskeleton, combining them in a Markovian estimation model in order to get the best information from each sensor. In order to demonstrate the effectiveness of our approach, simulation results regarding a set of human footsteps, with four IMUs and three encoders attached to the lower limb exoskeleton, are presented. A comparative study between the Markovian estimation system and the standard one is performed considering a wide range of parametric uncertainties.

Research and development on robotic devices for rehabilitation and assistance have received much attention in the last few years. Alternative designs of exoskeletons for upper and lower limbs have been proposed in the literature (e.g., [

As the control approaches for this class of robots are generally performed based on absolute and relative angles of their segments, they depend on reliable and accurate position measurements. In this sense, inertial measurement units (IMUs) have been successfully applied to gait identification (e.g., [

In [

In order to deal with the performance limitations of lower limb exoskeletons, in [

It is known that when the dynamic acceleration increases, the number of reliable accelerometers reduces at a certain instant of time. Consequently, in some segments, like the trunk and foot, the high incidence of dynamic accelerations can degrade the angular estimation. Furthermore, human walking has great variability and unpredictability related with the speed and length of each step, which cause different patterns of dynamic accelerations in each segment. In order to deal with this problem, we propose a model that encompasses the influence of whole dynamic accelerations in a lower limb exoskeleton.

This model is developed based on Markovian systems, which are a particular class of randomly time varying-systems. The main feature of this approach is that the probability of the system being in a different configuration in the next time instant depends only on the present system configuration, regardless of the time instant considered. Markovian modeling has been used to solve control problems in robotic systems. For example, in [_{∞} controller was implemented in an underactuated manipulator, subject to abrupt changes on its configuration. It is conceived of to deal with a set of joint faults.

In this paper, we propose a new approach based on the Markovian model to estimate angular positions in lower limbs exoskeleton to overcome the accelerometer reliability problem over all gait phases. In contrast to [

In order to show the effectiveness of the estimation model proposed, we present simulations based on a set of human footsteps, with four IMUs and three encoders attached to the lower limb exoskeleton. A comparative study between estimations based on Markovian and standard Kalman filters is performed, considering a wide range of parametric uncertainties.

This paper is organized as follows: Section 2 presents the accelerometer and gyroscope models, the standard sensor fusion strategy and the proposed Markov Jump Linear Systems (MJLS)-based model. Section 3 presents the Markovian Kalman filter algorithm and the required conditions to select the best information from each sensor. Finally, Section 4 presents simulated results and discussions.

In this section, the Markovian model and the sensor fusion strategy designed to estimate the absolute and relative angular positions of exoskeletons for lower limbs are presented. The proposed approach will be applied to the exoskeleton, Exo-Kanguera, where rehabilitation protocols have been used to help stroke and spinal cord injured patients during walking rehabilitation. It is driven by a series of elastic actuators that allow the implementation of impedance control, ensuring patient safety and the ability to define different interaction behaviors.

In the following, gyroscope and accelerometer models are described. Standard sensor fusion approaches for an individual IMU and the proposed MJLS-based model, which includes signals of all IMUs attached to the exoskeleton, are presented.

The IMUs are attached to each segment of the body, as shown in _{g}(_{g}(_{g}(_{g} is the Markov process correlation time and _{b}_{g}(

Analogously, the acceleration of the link, _{a}(_{a}(_{a}(_{a} is the Markov process correlation time and _{b}_{a}(

Since the variances,
_{g} and _{a} can be generated considering a set of curves for

Standard sensor fusions based on Kalman filters take into account the gyroscope signal as the main source for position estimation. They use the accelerometer signal as a redundant information to correct the estimated value from the gyroscope, usually corrupted by integration errors. The estimated angular velocity and bias from the gyroscope are given by:

Taking the difference between real and estimated values, we obtain:

Hence, the state space representation of the estimation system is obtained as:
_{g}(

The angular position obtained from the accelerometer is given by:
_{e}_{a}, is assumed to contain only the component of the gravitational acceleration in the

The output equation of the state space representation is given by the difference between the estimated position from the accelerometer and estimated position from the gyroscope, that is,

Therefore,

The final estimated values for the angular position and gyroscope bias are given by:
_{g} are the estimated values of the angular position and offset errors, respectively. These errors are estimated by the Kalman filter using the sensor fusion approach.

In this section, we propose a new MJLS model for angular position estimation of lower limb exoskeletons. Considering one leg of the exoskeleton, Exo-Kanguera (see

Absolute angles: _{B}_{T}_{S}_{F}

Relative angles: _{h}_{k}_{a}

The spatial configuration of the exoskeleton, Exo-Kanguera, at a given instant of time, can be obtained by estimating one absolute angle and three relative angles. The absolute angle is obtained by the sensor fusion approach presented in Section 2.2. The relative angles are measured by encoders coupled to the joints of the exoskeleton.

The MJLS model for angular position estimation of lower limbs exoskeletons can be described by the following state-space equations:
^{n}^{m}^{q}^{q}_{i}_{i}_{g}_{i}_{i}_{g}_{i}_{i}_{IMU}_{aIMU}) and the estimates calculated by the gyroscopes (_{gIMU}); and Δ_{j}

Considering the gyroscope, accelerometer and gyroscope bias models, matrices _{i}_{IMU}

In the proposed approach, only one inertial sensor is used at each instant in combination with three relative sensors. As the system has four inertial sensors, the Markovian model presents four states, as shown in

In _{B}_{T}_{S}_{F}

After generating the estimation errors for all angles from the Markovian filter, the estimated absolute angles are given by:

In this section, we present the Markovian Kalman filter (MKF) and the guidelines to choose the Markovian states for each instant of time. Consider the discrete-time version of the Markovian model, described in Section 2.3, given by:
^{−1}(Φ − _{k}_{k}_{+1} is the Kalman filter (KF), since all operation modes are known at time

Predicting equations:

Updating equations:

On the other hand, the weighting matrix,

The reliability of the accelerometer signal is essential to define the current Markovian state, in order to be used in the Kalman filter. When the module of the triaxial accelerometer vector is near the module of the gravity vector, there exists a low incidence of dynamic accelerations. We can define that the current incidence of dynamic accelerations and the Markovian state are chosen as:
_{i}

Thereby, the general condition to verify the reliability of the current IMU reading is given by:

Let the magnitude vector measured by the four IMUs attached in a lower limbs exoskeleton (see

Then, applying

Notice that we have two possible alternatives:

Reliable measurement signal: In this case, the proposed MKF performs the prediction and update of the estimated angle,

Unreliable measurement signal: In this case, the MKF performs only the prediction,

In order to present the simulation results, we consider the following assumptions:

The angular estimate is performed for one leg of the exoskeleton (trunk to foot; see

In the standard case, modeled in Section 2.2, the system is replicated for each segment of the exoskeleton. More details on the implementation of the standard Kalman filter can be seen in [

In the Markovian case, modeled in Section 2.3, all segments are included in a unified model.

For both estimate approaches, the weighting matrices parameters are given in

As discussed in Section 3, to define the sequence of Markov jumps among B, T, S and F,

Notice in

In order to verify the effectiveness of the filters when the system is subject to parametric uncertainties, both estimators were run, varying the components of the system matrices. Variations from 0% to 20% are assumed in the nominal values. Furthermore, 100 Monte Carlo simulations were performed. In

A MJLS-based position estimation approach for lower limbs exoskeletons is presented in this paper. Unlike the standard position estimations, where inertial sensors are designed to work individually for each link, the proposed approach considers a collective modeling of all inertial sensors attached to the exoskeleton. They are combined in a Markovian estimation model. Simulations results were presented considering the kinematic model of an exoskeleton for lower limbs, where four IMUs and three encoders were used. The results show the advantage of the proposed Markovian estimation model if compared with the standard one, even when they are subject to a wide range of parametric uncertainties.

This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Apoio à Pesquisa do Estado de São Paulo (FAPESP), under grant 2012/05552-9.

The authors declare no conflict of interest.

_{∞}controls of underactuated manipulators

The end of the swing phase of a step through the Exo-Kanguera.

Inertial measurement units (IMUs) attached to the body

Markovian state jumps and model. (

Markovian chain.

Body/trunk segment. (

Thigh segment. (

Shank segment. (

Foot segment. (

Filter effectiveness.

Table of Markovian states.

_{B} |
_{T} |
_{S} |
_{F} | |
---|---|---|---|---|

B | 1 | 0 | 0 | 0 |

T | 0 | 1 | 0 | 0 |

S | 0 | 0 | 1 | 0 |

F | 0 | 0 | 0 | 1 |

System parameters.

_{g} |
_{b}_{g} |
_{a} |
_{g} | ||
---|---|---|---|---|---|

Body/Trunk | 0.01 | 0.0103 | 0.0003 | 46.562 | 839.70 |

Thigh | 0.01 | 0.0017 | 0.0005 | 329.08 | 2138.5 |

Shank | 0.01 | 0.0008 | 9.6405 | 100.94 | 1465.3 |

Foot | 0.01 | 7.7482 | 18.594 | 291.29 | 3567.5 |