3.1. Inverse geometric model

The design of the structure was carried out to minimize the coupling of the joint coordinates of the structure through the application of an architecture based on the “multipteron” family [Reference Gosselin, Laliberté and Veillette25, 26], keeping only the sagittal movement of the pelvis mechanically restricted due to its low influence on the rehabilitation process of human gait [13, Reference Ohnuma and Lee23]. For this, we propose the application of a structure defined as 4-PRRU + PRRS as a form of “pentapteron,” as shown in Fig. 2, where $P$ is an actuated prismatic joint, $R$ represents a rotational joint, $U$ is a universal joint, and $S$ represents a spherical joint. Point P of the mobile platform is the point where a seat for the patient will be installed.

Figure 2.

(a) Schematic drawing for the structure of the active support of the pelvis, based on the “pentapteron” (4-PRRU + PRRS); and (b) position of the point of interest P in relation to the vertices of the mobile platform.

Figure 3.

Analysis of the position of point A in relation to the inertial frame and leg 1.

In Fig. 2, translational and rotational joints have been grouped into cylindrical joints $q_{i},i=1to5$ , to facilitate representation in the scheme. However, translation and rotation joints were used during the construction of the prototype described in Section 4. The universal joints are represented by cross-cylinders, and the spherical joint as a dashed circumference placed behind the mobile platform, Fig. 2.

For the modeling of this system, equipollent coordinates [Reference Selig27] were used for the references illustrated in Fig. 2(a) so that each joint coordinate $q_{i}$ is in one of the directions of the unit vectors of the frame of reference. Therefore, for each leg $i$ , the vector $u_{i}$ that connects the active joint of this leg to one of the points on the mobile platform that belongs to a plane perpendicular to the unit vector of the actuator direction is highlighted. Based on these facts, we proceed with the modeling of this structure, analyzing each leg $i$ individually and systematizing the equations obtained in a matrix relating to the joint and operational coordinates.

To analyze the position of point A of the mobile platform from a leg $i$ in relation to the inertial reference can be equated with the position in relation to the references of the leg in question. For the case of leg 1, this relationship is illustrated in Fig. 3 and formalized in (1).

(1) \begin{equation} \,^{\mathbf{0}}\,\mathbf{\!r}_{\mathbf{p}}+\mathbf{r}_{\mathbf{A}}= {}^{\mathbf{0}}\mathbf{r}_{\mathbf{1}}+q_{1}\mathbf{i}_{\mathbf{1}}+\mathbf{u}_{\mathbf{1}} \end{equation}

where $^{\textbf{0}}{\textbf{r}}_{\textbf{p}}$ is the position of point P with respect to $X_{0}Y_{0}Z_{0}, \textbf{r}_{\textbf{A}}$ is the relative position of point A with respect to point $P$ , and $\textbf{i}_{\textbf{1}}$ is the unit vector of the X direction of reference frame $X_{1}Y_{1}Z_{1}$ . The vector u1 belongs to a plane parallel to the plane $Y_{1}Z_{1}$ , therefore perpendicular to the unit vector $\textbf{i}_{\textbf{1}}$ , and that $\textbf{i}_{\textbf{1}}$ = $\textbf{i}_{\textbf{0}}$ due to the fact of adopting references with equipollent coordinates. Therefore, when performing a dot product on both sides of (1) by $\textbf{i}_{\textbf{0}}$ :

(2) \begin{equation} q_{1}=\left({}^{\mathbf{0}}\mathbf{r}_{\mathbf{p}}+\mathbf{r}_{\mathbf{A}}- {}^{\mathbf{0}}\mathbf{r}_{\mathbf{1}}\right)\cdot \mathbf{i}_{\mathbf{0}}=\left(\left\{^{0}{r}_{p_{x}}^{0}{r}_{p_{y}}^{0}r_{{p_{z}}}\right\}+\left\{r_{{A_{x}}}r_{{A_{y}}}r_{Az}\right\}\!-\!\left\{-\frac{dx}{2}\frac{dy}{2}0\right\}\right)\!\cdot\! 100\Rightarrow q_{1}= {}^{0}r_{{p_{x}}}+\frac{dx}{2}+r_{A}\cos \phi \end{equation}

where $\phi$ corresponds to the angle between the vectors $\textbf{r}_{\textbf{A}}$ and $\textbf{i}_{\textbf{0}}$ , as illustrated in Fig. 4, which is also defined as the orientation of the mobile platform in terms of its rotation around the $Z_{0}$ axis.

Figure 4.

Angles between the unit vectors and the position vectors of the ends in relation to point P: (a) angles of rotation around X ₀; and (b) rotation around Z ₀.

The analogous procedure is made to legs 2 to 5 to obtain the following equations:

(3) \begin{equation} q_{2}=\mathbf{(}^{\mathbf{0}}\mathbf{r}_{\mathbf{p}}+\mathbf{r}_{\mathbf{B}}- {}^{\mathbf{0}}\mathbf{r}_{\mathbf{1}}\mathbf{)}\cdot \mathbf{k}_{\mathbf{0}}\mathbf{q}_{\mathbf{2}}= {}^{0}\!r_{{p_{z}}}+r_{B}\cos \theta \end{equation}

where angle $\theta$ corresponds to the angle between the unit vector $\textbf{k}_{\textbf{0}}$ and the vector $\textbf{r}_{\textbf{B}}$ , as illustrated in Fig. 4.

(4) \begin{equation} q_{3}=\,^{0}r_{py}+r_{C}\cos\! \left(\theta +\psi \right)+\frac{dy}{2} \end{equation}

where angle $\phi +\psi$ represents the orientation around the $X_{0}$ axis of the mobile platform, as shown in Fig. 4.

(5) \begin{equation} q_{4}=\,^{0}r_{pz}+r_{D}\;cos(\theta +\phi ) \end{equation}

where angle $\theta +\varphi$ is the angular difference between vector $\textbf{r}_{\textbf{D}}$ and unit vector $\textbf{k}_{\textbf{0}}$ , as defined in Fig. 4.

(6) \begin{equation} q_{5}= {}^{0}r_{px}-r_{E}\cos\! \left(\varphi \right)+\frac{dx}{2} \end{equation}

Equations (2) to (6) present the relations of the operational coordinates $^{\textbf{0}}\textbf{r}_{\textbf{p}}, \theta$ , and $\varphi$ with the joint coordinates $q_{1}, q_{2}, q_{3}, q_{4}$ , and $q_{5}$ . Through these, it is possible to find the Jacobian matrix of the structure, which is used to determine the possible singular positions.

3.2. Singularity analysis

First, it is necessary calculate the time derivative of (2) to (6). Their respective time derivatives are presented in (7a) to (7e) and the matrix form, $[J_{q}]\{\dot{q}\}=[J_{x}]\{\dot{x}\}$ , that groups these equations is presented in (7f):

(7a) \begin{equation} \dot{q}_{1}= \dot{\,^{0}r_{px}}-r_{A}\sin\!\left(\varphi \right)\!\dot{\varphi } \end{equation}

(7b) \begin{equation} \dot{q}_{2}= \dot{\,^{0}r_{pz}}-r_{B}\sin\!\left(\theta \right)\!\dot{\theta } \end{equation}

(7c) \begin{equation} \dot{q}_{3}= \dot{\,^{0}r_{py}}-r_{C}\sin\!\left(\theta +\psi \right)\!\dot{\psi } \end{equation}

(7d) \begin{equation} \dot{q}_{4}= \dot{\,^{0}r_{pz}}-r_{D}\sin\! \left(\theta +\phi \right)\!\dot{\theta } \end{equation}

(7e) \begin{equation} \dot{q}_{5}= \dot{\,^{0}r_{px}}+r_{E}\sin\! \left(\varphi \right)\!\dot{\varphi } \end{equation}

(7f) \begin{equation} \left[J_{q}\right]\left\{\dot{q}\right\}=\left[J_{x}\right]\left\{\dot{x}\right\}\Rightarrow \left\{\begin{array}{c} \dot{q}_{1}\\[5pt] \dot{q}_{2}\\[5pt] \dot{q}_{3}\\[5pt] \begin{array}{c} \dot{q}_{4}\\[5pt] \dot{q}_{5}\\[5pt] \end{array}\,\\[5pt] \end{array}\right\}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & -r_{A}\;\textrm{sin}\!\left(\varphi \right)\\[5pt] 0 & 0 & 1 & -r_{B}\sin\!\left(\theta \right) & 0\\[5pt] 0 & 1 & 0 & -r_{C}\sin\!\left(\theta +\psi \right) & 0\\[5pt] 0 & 0 & 1 & -r_{D}\sin\!\left(\theta +\phi \right) & 0\\[5pt] 1 & 0 & 0 & 0 & r_{E}\sin\!\left(\varphi \right) \end{array}\right] \left\{\begin{array}{c} {}^{0}{\dot{r}_{{P_{x}}}} \\[5pt] {}^{0}{\dot{r}_{{P_{y}}}} \\[5pt] {}^{0}\dot{r}_{{P_{z}}} \\[5pt] \dot{\theta } \\[5pt] \dot{\varphi } \end{array} \right\} \end{equation}

In this way, it is possible to notice that $[J_{q}]=[I]$ , which means that the singular positions will exist only for the case where $\det\! ([J_{x}])=0$ . Therefore, these cases can be expressed according to (8):

\begin{equation*} \det\! \left(\left[J_{x}\right]\right)=r_{A}r_{D}\sin\!\left(\phi +\theta \right)\sin\!\left(\varphi \right)-r_{B}r_{E}\sin\!\left(\theta \right)\sin\!\left(\varphi \right)- \end{equation*}

\begin{equation*} -r_{A}r_{B}\sin\! \left(\theta \right)\sin\! \left(\varphi \right)+r_{D}r_{E}\sin\! \left(\phi +\theta \right)\sin\! \left(\varphi \right)=0 \end{equation*}

(8) \begin{equation} \Rightarrow r_{D}\sin\! \left(\phi +\theta \right)=r_{B}\sin\! \left(\theta \right) \end{equation}

Figure 5.

(a) Prototype of the motor module and (b) schematic drawing of the mechanism.

Therefore, whenever (8) is true, the system will be in a position of singularity and will be subject to loss of control and/or problem of mobility in this position. Based on this information, to design this structure it is necessary to guarantee that the selected dimensions of $r_{B}, r_{D}$ , and $\phi$ do not allow the occurrence of (8) at no point in the workspace at $\theta$ . It is also possible to notice that the translational degrees of freedom of this structure are limited only by the respective limits of the workspace, since these do not present terms in the Jacobian matrix $[J_{x}]$ .

To reduce the cost of high-speed linear actuators, an actuator model was proposed, based on 24-V DC motors coupled to rotary encoders, and the angular movement of these motors was transformed into linear displacement through a 3-bar crank-rod mechanism type. The scheme designed for this system is represented in Fig. 5, and the relationship between $q_{i}$ and $\theta _{i}$ is described in (9).

(9) \begin{equation} q_{i}=r_{2i}\cos \theta _{i}+\sqrt{{r_{3i}}^{2}-{r_{2i}}^{2}\;sin^{2}\theta _{i}} \end{equation}

3.3. Dynamic model and simulation optimization

To define the dimensions of the mechanical elements and actuators, a computational model considering geometric and dynamic characteristics was developed. The patient’s mass was considered as punctual dynamic loading generated by it on the linear guides $q_{i}$ and transmitted to the DC motors $M_{i}$ through the connecting crank-rod mechanism. The approach is valid due to the low speeds and accelerations present in the kinematics of human gait rehabilitation [Reference Gonçalves and Krebs28]. Therefore, the mass and inertia of the bars and other mechanical elements present in this module can be disregarded without prejudice to the system specifications. A critical mass $M_{P}=150\,{\rm Kg}$ was considered.

The movement requirements were established according to the human gait with a focus on the sacroiliac joint, as shown in Table V.

For the calculation of the dynamic model, it was considered: bars that connect the module’s mobile platform to the linear guides must withstand the maximum static load, specified as the critical mass of the system; linear guides must traverse the specified workspace within the critical cycle time ( $t_{c}=0.4\,{\rm s}$ ); bars that connect the linear guides to the DC motors must punctually support the maximum dynamic load, disregarding the effect of inertia to simplify the model; actuators 2 and 4 will be implemented together with a passive spring system to relieve the patient’s static load, as shown in Fig. 6, allowing actuators to focus only on dynamic efforts; and an additional spring has been connected to the mobile platform, as shown in Fig. 6, without prejudice the defined workspace, to improve the passive support of the structure and relieve the static load of the other actuators.

Figure 6.

(a) CAD version and (b) the built prototype.

Therefore, from the proposed directives, it is defined that the linear guides must be able to start their respective trajectories from the beginning of the course with zero initial velocity and finish the movement in the final position again with zero velocity. The dynamic equation that defines the greatest effort $F_{i}$ applied under each actuator $q_{i}$ can be defined as:

(10) \begin{equation} \mathbf{F}_{\mathbf{i}}=\mathbf{f}_{\mathbf{r\,i}}\mathbf{M}_{\mathbf{P}}\frac{\mathbf{d}_{\mathbf{i}}}{{\mathbf{t}}_{\mathbf{c}}^{\mathbf{2}}}\quad \mathbf{i}\;\mathbf{=}\;\mathbf{1}\mathbf{,}\,\mathbf{2}\mathbf{,}\,\mathbf{4}\mathbf{,}\,\mathbf{5}\Rightarrow \mathbf{f}_{{\mathbf{r}_{\mathbf{i}}}}\;\mathbf{=}\;\mathbf{0}\mathbf{.}\mathbf{5}\,\,\mathbf{i}\;\mathbf{=}\;\mathbf{3}\Rightarrow \mathbf{f}_{\mathbf{r\,i}}\;\mathbf{=}\;\mathbf{1}\,\, \end{equation}

where $d_{i}$ corresponds to the working space specified for each linear guide rail, $t_{c}$ is the time to complete the gait cycle considered as $0.4\,{\rm s}$ , and the parameter $f_{ri}$ corresponds to a correction factor applied in the equation to consider the weight distribution between actuators 1 and 5, and actuators 2 and 4. The actuator 3 acts alone in the $j_{0}$ direction, which means that it does not need correction, and therefore $f_{r3}=1$ . The model also considers the minimum requirements for electrical power and angular speed of the specified DC motors to allow the correct movement of the connecting crank-rod mechanism. A free body diagram was constructed showing the considered efforts, as shown in Fig. 7. In this diagram, $T_{i}$ represents the input angular momentum of the motor $M_{i}$ , and $F_{i}$ is the force applied on $q_{i}$ .

Figure 7.

Free body diagram of the connecting crank-rod mechanism of the actuators.

Applying the equilibrium moment conditions to the free body diagram in Fig. 7, with respect to the point $E_{i}$ , the moment $T_{i}$ can be written as:

(11) \begin{equation} T_{i}=F_{i}r_{2i}\;sin\,\theta _{i} \end{equation}

differentiating (9) in time and isolating $\dot{\theta }_{i}$ is obtained:

(12) \begin{equation} \dot{\theta }_{i}=\frac{\dot{q}_{i}}{-r_{2i}\;sin\,\theta _{i}-\frac{{r_{2i}}^{2}\;sin\,\theta _{i}\;cos\,\theta _{i}}{\sqrt{{r_{3i}}^{2}-{r_{2i}}^{2}\;sin^{2}\,\theta _{i}}}} \end{equation}

Therefore, to obtain the expression for the minimum power required for each electric motor is applied (13):

(13) \begin{equation} P_{i}\left[W\right]=\frac{\theta _{i}\tau _{i}}{9.55} \end{equation}

To calculus the tension applied to the cross sections of each bar, was used as a criterion for the comparison $\sigma _{ij}$ of the stress calculated in each cross section through the theory of strength of materials [Reference Uicker, Pennock, Shigley and McCarthy29], according to (14), where $F_{j}$ are the efforts applied to the element, $r_{ij}$ is the analyzed bar, and $I_{ij}$ is the moment of inertia of the bar:

(14) \begin{equation} \sigma _{ij}=\frac{F_{j}r_{ij}}{4I_{ij}} \end{equation}

To obtain the optimal dimensions of the bars that will compose the mechanism, the tension calculated according to (14) is compared with the yield strength of the selected material defined as aluminum ( $\sigma _{max}=180\,{\rm MPa}$ ). To facilitate the bar specification process and reduce manufacturing costs, all bars in the system were considered as rectangular tubes with cross sections compatible with the materials available. The factor of safety equal two was used. All the requirements were combined in a computational model developed in MATLAB® together with a differential evolution algorithm forming an interactive simulation optimization system. The simulation optimization parameters are number of replications = 10, population size = 50, maximum iterations = 200, crossover rate = 0.8, and mutation probability = 0.005.

The results obtained through this process are shown in Table I. The bars $r_{2i}$ and $r_{3i}$ are the ones that make up the connecting crank-rod mechanism of each actuator $i$ , being all similar to each other, as shown in Fig. 5. The bars $r_{4i}$ and $r_{5i}$ represent the connection bars between the linear guide rail and the mobile platform for each leg $i$ , as shown in Fig. 1.

The simulation also determined the necessary characteristics for the DC motors, as shown in Table II, as described in (12) and (13).

7.1. Movements of the novel active BWS

The proposed movements of the novel active BWS are presented in Fig. 14. The measurements of the mobile platform were performed using a digital inclinometer (resolution of 0.1°) to define the actual workspace of the structure and compare it with that specified in the mathematical model. In this experiment, 10 repetitions were also performed for each measured angular degree of freedom. In the case of linear degrees of freedom, the nominal measures of the linear guides applied to the structure were used as the workspace. The results are presented in Table V.

Figure 14.

Movement tests of the mobile platform of the novel BWSS prototype. (a) Rotation in the transverse plane (about the OZ axis); (b) rotation in the frontal plane (around the OX axis); (c) translation on the OX axis; (d) translation on the OY axis; and (e) translation on the OZ axis.

From Table V, it is possible to notice that the workspace obtained in the prototype meets the proposed specifications considering the mobility of the pelvic joint. It is also noted that the selected design also allows angular amplitudes above the minimum, which will allow the structure to be used to perform controlled movements for balance training.

A comparison was made with the points obtained for one of the motors of the structure about the same movement performed by the CAD/CAE model developed [Reference Gonçalves and Rodrigues36]. In this experiment, a simulation of the mechanical system was performed moving 150 mm in 0.4 s, and the points obtained were saved. Afterward, the same movement was performed 10 times on one of the structure’s motors, and the telemetry was collected. The comparison between the simulated and experimental results is represented in Fig. 15.

7.2. Verification of control requirements

The test carried out sought to validate whether the use of the differential evolution algorithm was able to obtain the constants $K_{P}, K_{I}$ , and $K_{D}$ capable of meeting the requirements proposed in Section 5.

The experiment carried out consisted of saving the data recorded in the input, output, and time Arduino when performing a displacement of 200 pulses and analyzing them using open Data Science libraries in Python and Jupyter Notebooks. The results obtained are visually represented through histograms as shown in Fig. 16.

Table V.

The amplitude of the mobile platform measured experimentally.

Figure 15.

Comparison between the experimental data measured in one of the prototype actuators and the data simulated in CAD.

Figure 16.

Histogram representing the distribution of the results obtained in the 11 repetitions of the control requirements validation experiment. From top to bottom, Settling Time Analysis Results, Percent Stationary Error, and Overshoot.

Through the visual analysis of the collected data, it was possible to notice that the specified accommodation time was consistently met, ensuring that the system presents an adequate response. Although the stationary error has been specified as null and the mechanical characteristics of the system guarantee a transfer function with Type Number 1, the error cannot be eliminated. This error will need to be investigated in detail in future works and may be linked to the limitations of the microcontroller. However, considering the constructive aspects of the structure, the stationary error of the structure can still be considered tolerable and not cause significant problems in the movement of the platform, guaranteeing the functionality of the equipment. In addition, the system did not show overshoot in any of the samples. This also ensures that frame-specified movements will not expose patients to an excessive range of motion and will prevent vibrations when reaching the desired position.

Another experiment carried out sought to validate the relationship between the number of pulses measured in the encoders. To carry out this experiment, a displacement of 200 pulses was applied to each of the motors with 10 repetitions, and the displacement of the linear cursor was measured with the aid of a precision ruler (resolution of 0.5 mm) fixed below the cursor rail. The results obtained are graphically represented in Fig. 17.

Figure 17.

Boxplot graph of the resolution calculated for each actuator, in pulses/mm.

It is noteworthy that as one of the actuators used an encoder with 500 pulses per revolution, it obtained a lower resolution than the others. The resolution presented a considerable variance in all cases which is connected with the mechanical and electronic limitations of the prototype. However, it was possible to observe that the resolutions remain close to each other in actuators with 600-pulse encoders, and within the expected value according to the mathematical model obtained in Section 3. With an estimated value for the resolution, it was also possible to estimate the maximum linear velocity reached by the modules, equal to approximately 1.29 m/s.

Therefore, based on the results obtained in the experiments, it can be concluded that the structure is valid for the practical tests foreseen with healthy patients.

7.3. Pilot study

Experimental trials have been approved by the ethics committee on human research at the Federal University of Uberlândia (CAAE 01305318.2.0000.5152). The tests were made using the proposed novel BWS in combination with the developed serious game and a treadmill. All participants gave their informed consent before enrollment. The trial was carried out on two healthy volunteers over 18 years of age, one male and the other female. The proposed objectives for the trial are as follows: check the operation of the structure together with the treadmill in operation, with the human gait running; check the control of the position of the main character of the game, during the execution of the human gait; evaluate gameplay and difficulty during actual use; and identify electrical and/or mechanical failures during operation. Two test sessions lasting about 5 min were performed. A photograph of one test session is shown in Fig. 18.

Figure 18.

Test session with the healthy participant.

The neutral position of the platform was adjusted according to the height of each participant. Then, the participants were accommodated on the structure, and the treadmill was turned on. Participants walked on the treadmill for a few minutes without the game running to check if the speed was adequate, and the platform was comfortable. After confirming the initial conditions with the participants, the game was started and the test session was video-recorded. At the end of the session, the recordings were interrupted and the volunteers presented their main impressions of the structure and the serious game.

The first feedback presented by the participants was about the execution of the gait on the treadmill with the platform. Both participants reported being able to walk normally, with little or no influence of the saddle on the movements performed. During the game, participants reported little difficulties in properly moving the character just moving to the left or right of the treadmill. The main causes of this problem were the limited space for lateral displacement. As for the gameplay of the game, the participants contributed to the correction of the position of some obstacles and the bonus cube. In the game event of platform inclination, the participants reported that the displacement was noticeable, being necessary to adjust the gait movements to the imposed position. The video footage of the experimental tests and the computational simulations presented in this paper can be accessed in: https://drive.google.com/drive/folders/17bZKsnUAPb1jSSVaHco1ahpjLiTBqhZM?usp=sharing