Experimental results are described in three subsections. The first one deals with the mechanical–electronic construction of the robotic standing wheelchair and presents a brief description of the prototype created in order to run experimental tests. Secondly, we validated the dynamic model of the human–robot system, through experimental identification and validation tests. Thirdly, the validation of the proposed control scheme is carried out through experimental tests supported by graphs showing the effectiveness of the control scheme.
4.1. Robotic Standing Wheelchair Construction
This work uses a non-holonomic robotic standing wheelchair, which was developed by the ARSI Research Group of the University of the Armed Forces ESPE. The standing wheelchair has two wheels driven by two DC motors independently (rear traction), in order to move the chair on a planar horizontal surface considering the non-holonomic restriction (Equation (5)). Two passive wheels (pivoting wheels) are located in the front part of the central axis to give greater stability to the robotic standing wheelchair; and an independent linear drive that allows the change from sitting position to standing position, by means of a DC motor. The position and relative orientation of the standing wheelchair can be known by means of the encoders installed on each of the motor shafts. The construction of the mechanical parts of the robotic system have been designed to fit together, resulting in an appropriate analysis of the center of mass and weight distribution, allowing the mobile platform and the standing joint to be unified as a single system as shown in
Figure 7.
The standing wheelchair robotic system was designed in such a way that the electronic components can be interconnected between the control elements and equipment, power, and energy supply. The system consists of DC motors, overcurrent prevention elements, an electronic control board, a computer, a peripheral extender, and a battery. The distribution of the elements, together with the communication links among them, is shown in
Figure 8. We describe below the different parts of the whole system.
(i) Mobile platform system: this section consists of two direct current motor that are controlled by a Roboteq card (Roboteq Inc., Scottsdale, AZ, USA), which incorporates PID controllers through a refeed with encoders of 400 pulses per revolution (velocities are subsequently transformed to rad/s). The motor controller card, through Rs232 serial communication, sends the actual velocity and position of the mobile platform to a computer, whereas the computer sends the maneuverability velocities for the control of the mobile platform (maneuverability velocities obtained by implementing a control algorithm). The PID control implemented in one of the motors is shown in
Figure 9, where it is observed that the velocity error tends to zero asymptotically. The Haalman method was considered for the tuning of the PID controllers, and the following parameters were obtained:
,
and
.
(ii) the standing system has an Atmel microcontroller (Atmel Corporation, San José, CA, USA) and a non-commercial control board as a processor, which was designed to satisfy the communication and processing requirements for the correct operation of the standing axis. The standing section consists of a direct current motor. The motor incorporates mechanical velocity reducers and a ball screw coupling, in order to generate a linear movement that allows the wheelchair to move on the
axis. In addition, the motor has an encoder attached to implement an internal loop PID controller. On the other hand, the standing motor control card (through RS232 serial communication) sends the standing angle and the motor velocity to the computer. On the other hand, the computer sends the maneuverability velocity commands for the standing control of the wheelchair (maneuverability velocity obtained by implementing a control algorithm); (iii) the computer has enough resources to process high-level programs. In the computer, the necessary calculations for the implementation of the control algorithms are carried out through the Windows operating system using mathematical software; (iv) electronic control board: its function is to distribute and regulate the power voltage of the motors and the computer system. In addition, it consists of voltage and current measurement elements in order to emit warning signals in the event of possible failures, discharges or disconnection of devices. A charging and connection status screen is integrated into the dash; (v) peripheral ports, which are responsible for communicating external devices (cameras, memory cards, etc.) with the internal computer; finally, vi) the battery supplies the necessary power to the system, which delivers up to 75 [A/h] with 12 V.
4.2. Dynamic Model Identification and Validation
The identification and validation of the dynamic parameters of the mathematical model (Equation (36) that represents the dynamics of the standing human–wheelchair system is tested in this subsection. The main objective of this process is to experimentally determine the numerical values of
with
, so that the dynamic model can be used in advanced control algorithms. The identification of dynamic parameters is the way to establish a relationship between the real results and the mathematical model developed, allowing to refine the model obtained until the behavior of the chair–user system shows sufficient precision to meet the requirements of the objectives of the desired control [
17,
20,
34].
In order to identify the dynamic parameters of the human–wheelchair system, Property 4 of the dynamic model described in
Section 2.2.1 is considered. For this, in order to estimate the acceleration of the robotic system, a first order filter is applied to Equation (37), obtaining:
Rewriting Equation (71) in a compact form, we obtain that:
where
is the Laplace transform variable, and
represents a positive fit constant. To estimate the parameters that best fit the dynamic model of the human–wheelchair system, the method of least squares is implemented. Therefore, the following expression is considered,
where
represents the values of the calculated dynamic parameters;
is a matrix that considers the variation of the dynamic model at any instant of time in which the dynamics of the real robotic system was excited; and
is the vector that incorporates the input excitation signals of the real robotic system.
Table 2 shows the dynamic parameters of the standing human–wheelchair system, considering a person of 75 kg mounted on the wheelchair and moving on a wooden surface.
Remark 3: The dynamic parameters presented in
Table 2 may vary according to the weight of the person and the type of surface on which the standing wheelchair moves.
Figure 10 presents experimental data for the validation process, where it can be observed the adequacy of the proposed dynamic model.
The so-built standing wheelchair allows the user to raise the chair from a seated to a standing position. The mechanism to raise the chair is controlled by the control scheme proposed in this work.
Figure 11 shows the autonomous movement of the robotic standing wheelchair with a user of 75 kg.
4.3. Control Scheme Implementation
In order to obtain experimental results with the human–wheelchair system for the execution of autonomous tasks, a partially structured scenario was considered. All the experimental tests presented in this work use the wheelchair presented in the
Section 4.1. The robotic wheelchair considers linear velocity and angular velocity for mobile platforms as input signals. In addition, it has an angular velocity as input signal for standing control on the
axis. On the other hand, the standing wheelchair has as output signals the displacement and rotation
with relation to reference frame
. In addition, the output signals for the mobile platform were linear and angular velocities, whereas the output signal for the standing position was the angular velocity.
Several experiments on the motion control of the standing wheelchair system were performed in order to illustrate the performance of the proposed controller. The most representative results are presented in the next section. Each one of the experiments was executed with different control objectives. It should be clarified that all experiments were implemented considering the proposed control scheme in
Figure 4. The difference of the experiments is in the control law to be implemented in the kinematic controller; the control law is selected based on the desired task.
The parameters of the proposed control scheme were adjusted, as shown in
Table 3, for all the experiments. The sampling time was set to
.
(a) First experiment
We consider Equations (52) and (64) for the implementation of the kinematic control law. Equation (52) considers as desired values
. Therefore, the desired task of the human–robot system must be defined on the
plane (without considering the orientation) with respect to reference frame
. The desired task and initial conditions for the controller are defined in
Table 4 for the experiment.
The main results of the first experiment are illustrated in
Figure 12,
Figure 13,
Figure 14 and
Figure 15.
Figure 12 shows the stroboscopic movement of the standing human–wheelchair system, based on real data.
Figure 13 and
Figure 14 show that the control errors
and
, respectively, converge to values close to zero asymptotically. It should be noted that the kinematic controller fulfills the objective of the desired task, while the dynamic compensation controller compensates for the dynamics of the human–wheelchair system. In other words, they are two independent controllers with different control objectives. The control action of the standing wheelchair is shown in
Figure 15.
Control errors
and do not tend to zero, because these control states are not part of the desired task, therefore they are not considered in the proposed control law of the Equation (52).
(b) Second experiment
We consider for this experiment the implementation of the kinematic control law based on Equations (53) and (64), respectively. Equation (53) considers as desired values
, therefore, the desired task of the human–robot system must be defined on the
axis, considering the orientation respect to inertial reference frame
. The desired task and initial conditions for the controller are defined in
Table 5.
The desired task considers a stabilization point on the
axis and the desired orientation with respect to the
axis of the inertial frame
. The main results of the second experiment are shown in
Figure 16,
Figure 17,
Figure 18 and
Figure 19.
Figure 16 shows the stroboscopic movement of the standing human–wheelchair system, based on real data.
Figure 16a shows the standing wheelchair robot in the initial condition, whereas
Figure 16b shows the standing wheelchair robot in the desired position.
Figure 17 and
Figure 18 show that the control errors
and
, respectively, converge to values close to zero asymptotically.
Figure 19 depicts the control actions of the standing wheelchair robot.
Control errors
and do not tend to zero, because these control states are not part of the desired task, therefore they are not considered in the proposed control law of the Equation (53).
In previous experiments, the performance of the mobile platform controller expressed in the Equation (52) and the orientation and standing controller expressed in the Equation (53) were tested. Both controllers considered only two of the four desired states that a complex task may require.
(c) Third experiment
For these final experiments, the implementation of the kinematic control law is considered (Equations (55) and (64)). Equation (55) considers a unified control based on primary and secondary objectives. The main objectives considered
, whereas as secondary objectives,
was defined. It is important to mention that the secondary objectives will always be met whenever they do not conflict with the primary objectives. The desired task is defined as
with respect to inertial reference frame
. The desired task and initial conditions for the controller are defined in
Table 6 for the experiments.
TRAJECTORY 1. The main results of the third experiment are observable in
Figure 20,
Figure 21,
Figure 22 and
Figure 23.
Figure 20 presents the stroboscopic movement of the standing wheelchair system, based on real data from trajectory 1. It is shown that the controller above has an adequate performance.
Figure 21 and
Figure 22 show that the control errors
and
, respectively, are ultimately bounded close to zero. We observe in
Figure 21 that the errors tend to be zero when the robot is on the proposed trajectory.
Figure 23 shows the control actions of the standing wheelchair robot.
TRAJECTORY 2. The results of the experiment are illustrated in
Figure 24,
Figure 25,
Figure 26 and
Figure 27.
Figure 24 presents the stroboscopic movement of the standing human–wheelchair system, based on real data from trajectory 2.
Figure 25 and
Figure 26 show that the control errors are
and
, respectively, which are limited to values close to zero, i.e., achieving final characteristics errors
.
Figure 27 shows the control actions injected into the standing wheelchair robot during the experimental test. From the results obtained, the adequate performance of the proposed controller was verified.
TRAJECTORY 3. Finally, in order to evaluate the robustness of the proposed control scheme, an experimental test was performed with a 91 kg person on the bipedestation chair. The kinematic control law proposed in Equation (55) and the dynamic compensation proposed in Equation (64) were implemented. For the dynamic compensation, the dynamic parameters obtained for a 75 kg person were considered, as shown in
Table 2. The experimental test dealt with the follow-up of a trajectory that best excited the dynamics of the robotic system. The desired trajectory selected is described by
,
,
, and
, while, the initial conditions of the robotic system were defined as:
,
,
, and
.
The results of the final experiment are shown in
Figure 28,
Figure 29,
Figure 30 and
Figure 31. The stroboscopic movement of the human–wheelchair system, based on real experimental data, is shown in
Figure 28.
Figure 29 and
Figure 30 show that the control errors are
and
, respectively, which are limited to values close to zero, i.e., achieving final characteristics errors
. Finally,
Figure 31 shows the maneuverability velocities applied to the robotic systems.
The proper performance of the proposed controllers was verified through six experimental tests, using three different control laws (the controller design was presented in
Section 3.2). The adequate performance of the control scheme proposed for the experimental robot showed that the standing wheelchair is capable of following the desired trajectory, compensating the dynamic effects. The latter can be presented by a change in the user’s position when using the standing wheelchair robot, or due to irregularities in the contact surface.
It should be emphasized that the first five experiments were carried out with a 75 kg person mounted on the standing chair, whereas the sixth experiment was carried out with a 91 kg person. From the results obtained experimentally, it can be concluded that in all experiments, the control error converges to values close to zero. Therefore, the control error
will be bounded, provided that the control error
is bounded. The control error
is different from zero when experimental tests are performed with a person with a different weight than the weight used in the identification process of the dynamic parameters of the model. The same occurs when the surface on which the experimental tests are performed is different from the surface used in the identification of the dynamic parameters. However, the control error
will be bounded as a function of the velocity error value
. This analysis is supported by the results obtained and the robustness analysis described in
Section 3.6, specifically by Equation (70). Therefore, from the results obtained in this work, it is feasible to propose a control scheme with adaptive dynamic compensation for the human–wheelchair system.