Intelligent trajectory planner and generalised proportional integral control for two carts equipped with a red-green-blue depth sensor on a circular rail

Abstract

This paper introduces an intelligent trajectory generator and a generalised proportional integral (GPI) control law for the movement of two carts on a circular motorised rail. The objective is to continuously monitor a person performing physical rehabilitation exercises and moving freely inside the circle. Each cart is equipped with an red-green-blue depth (RGB-D) device so that the patient is monitored from two points of view. One RGB-D device is dedicated to track the body part being exercised, while the other focuses on the person’s face to detect his/her emotional state. The proposed feedback control scheme is based on an exact feed-forward action combined with a velocity vector structural estimation using a model-based integral state re-constructor. Furthermore, the proposed intelligent trajectory generator comprises the references required to move the carts based on the instantaneous position of the patient and the current position of both carts. It also establishes the role (master or slave) of each cart, the direction that each cart must follow, and it generates smooth trajectories that avoid collisions between the carts. Moreover, the paper introduces a description of the displacements of both carts and the positioning of the tilt and pan angles of each RGB-D sensor so as to track the patient’s face and the body from the best viewpoints. The performance of the intelligent trajectory generation and the control of the two carts were analysed with numerical simulations. A comparison was made between the developed GPI control and a standard proportional integral derivative (PID) control. The simulation results show the effectiveness of the trajectory planner, and demonstrate that the GPI control has a better dynamic response than the PID control as well as a better performance in terms of the integral squared tracking error, the integral absolute tracking error, and the integral time absolute tracking error.

Keywords

Physical rehabilitation facial emotion detection moving cart motorised circular rail RGB-D sensor GPI control

1. Introduction

The popularity of computer-based physical rehabilitation systems is constantly increasing. These rehabilitation systems typically use depth cameras to detect and track humans [1, 2, 3, 4, 5, 6]. In this sense, facial emotion detection of a patient performing the exercises helps to understand how he/she feels during the rehabilitation program [7]. This is of great importance in the treatment of certain diseases such as chronic pain as emotions play a major role during rehabilitation [8]. Facial analysis is also at the core of exercise-induced fatigue [9]. Emotional information is useful as feedback to the system to modify and adapt the exercise category and difficulty [10]. Our vision-based solutions are grounded on human detection [11, 12, 13] and tracking [14]. This paper is also inspired in previous research on multi-robotics [15, 16], tracking robotics [17] and rehabilitation robotics [18, 19].

When monitoring people undergoing physical rehabilitation exercises, it is mandatory to count on the best viewpoint of the exercised part of the body by modifying the angles of the camera accordingly [20, 21]. Moreover, if facial emotion is to be detected in order to know the mood derived from the rehabilitation exercises, then another viewpoint must be dedicated to capture the face of the patient. Red-green-blue depth (RGB-D) sensors must be smartly placed in the optimal positions to acquire images of the patient’s exercised part of the body and face. For this reason, mechanical solutions and control strategies are being developed for human rehabilitation exercises [22, 23, 24, 25]. In addition, as RGB-D images provide detailed depth information instead of specific 3D data acquired by traditional 3D sensors, they have fewer computation costs and are more suitable for real-time face expression recognition [26].

This paper, an extension of a recent conference paper [24], describes the behaviour of a motorised circular rail equipped with an intelligent trajectory generator which is capable of smartly relocating two carts, each one equipped with an RGB-D sensor, in order to monitor the patient’s face and the physically rehabilitated body part from two complementary views [27, 28]. The principal contributions of this research in regards to a recent one [24, 29] are (i) the design of an extension to the generalised proportional integral (GPI) control technique for both stabilisation and trajectory tracking tasks for each of the carts comprising the circular motorised platform, and (ii) the possibility that the patient moves freely inside the circle traced by the rail. In relation to the first contribution, the new control law has the advantage that no asymptotic observers or time discretisations are needed in the feedback loop to estimate the states commonly used in the design of traditional state-based feedback controllers for these systems [30, 31]. In addition, under the assumption that the patient can move freely, we define a trajectory generator for each of the two carts on the platform which work in a coordinated manner to monitor both desired parts of the body. At the same time, the security of the system is maintained and it is ensured that both carts never collide during their tracking of the person undergoing a rehabilitation program.

The paper is structured as follows. Section 2 presents a general description of the different elements comprising the circular motorised platform. Section 3 describes the dynamic model of the carts in charge of monitoring the patient, the dynamical model of the carts and the problem formulation. Section 4 introduces a complete description of the smart trajectory generator in charge of generating the reference trajectories of the carts, which are necessary to optimally locate the patient’s face and physical rehabilitation exercises. Section 5 presents the GPI control methodology applied to the stabilisation and tracking problem of the carts. In this section, it is proved that the GPI controller produces an asymptotically, exponentially convergent tracking error behaviour in relation to the origin of the coordinates in the error space. Section 6 introduces a series of numerical simulations to evaluate the performance of the trajectory generator and control. Lastly, the more relevant conclusions are drafted in Section 7.

2. General description of the system

As mentioned above, the ultimate practical application of this proposal is to monitor a patient performing a series of physical rehabilitation exercises from two viewpoints, in this way covering the face from one RGB-D camera and the part of the body being rehabilitated from another. In other words, in first place, the body part that is being exercised must be monitored to supervise that the exercises are being correctly performed. This will help to correct any bad movement. From the other cart facial expressions are detected in order to infer when the exercises are causing pain, dizziness, fatigue, etc. In this way, it will be possible to adapt the exercises to each patient. In order to carry out this process, a circular platform, composed of two carts (denoted as $Q$ and $S$ ), has been designed. Figure 1 provides a general view of the proposed circular motorised rail system showing the initial configuration of the carts and the final configuration of the carts after using the intelligent trajectory planner and Table 1 shows a description of the different symbols that will be introduced and described along this research work.

Table 1
List of symbols

Symbol	Description	Units
$\mathbf{P}(t)=[x_{p}(t),y_{p}(t),z_{p}(t)]^{T}$	Position of the patient in the circular rail	$[m]$
$\mathbf{P_{b}}(t)=[x_{b}(t),y_{b}(t),z_{b}(t)]^{T}$	Position of the body part to be monitored in the circular rail	$[m]$
$\mathbf{v_{p}}(t)=[x_{v}(t),y_{v}(t),z_{v}(t)]^{T}$	Direction towards which the patient is facing (principal vector)	-
$\mathbf{v_{s}}(t)=[x_{s}(t),y_{s}(t),z_{s}(t)]^{T}$	Direction to which the patient’s body part to be monitored is pointing (secondary vector)	-
$\mathbf{x_{Q}}(t)=[\beta_{Q}(t),\gamma_{Q}(t),\alpha_{Q}(t)]^{T}$	Configuration of cart Q in the circle formed by the rail	$[\textit{rad}]$
$\qquad\beta_{Q}(t)$	Angular position of cart Q around the circle formed by the rail	$[\textit{rad}]$
$\qquad\gamma_{Q}(t)$	Angular position of the pan of the RGB-D sensor mounted on cart Q	$[\textit{rad}]$
$\qquad\alpha_{Q}(t)$	Angular position of the tilt of the RGB-D sensor mounted on cart Q	$[\textit{rad}]$
$\mathbf{x_{S}}(t)=[\beta_{S}(t),\gamma_{S}(t),\alpha_{S}(t)]^{T}$	Configuration of cart S in the circle formed by the rail	$[\textit{rad}]$
$\qquad\beta_{S}(t)$	Angular position of cart S around the circle formed by the rail	$[\textit{rad}]$
$\qquad\gamma_{S}(t)$	Angular position of the pan of the RGB-D sensor mounted on cart S	$[\textit{rad}]$
$\qquad\alpha_{S}(t)$	Angular position of the tilt of the RGB-D sensor mounted on cart S	$[\textit{rad}]$
$\mathbf{x_{Q}^{}}(t)=[\beta_{Q}^{}(t),\gamma_{Q}^{}(t),\alpha_{Q}^{}(t)]^{T}$	Reference configuration of cart Q in the circle formed by the rail	$[\textit{rad}]$
$\qquad\beta_{Q}^{*}(t)$	Reference angular position of cart Q inside the circle	$[\textit{rad}]$
$\qquad\gamma_{Q}^{*}(t)$	Reference angular position of the pan of the RGB-D sensor on cart Q	$[\textit{rad}]$
$\qquad\alpha_{Q}^{*}(t)$	Reference angular position of the tilt of the RGB-D sensor on cart Q	$[\textit{rad}]$
$\mathbf{x_{S}^{}}(t)=[\beta_{S}^{}(t),\gamma_{S}^{}(t),\alpha_{S}^{}(t)]^{T}$	Reference configuration of cart S in the circle formed by the rail	$[\textit{rad}]$
$\qquad\beta_{S}^{*}(t)$	Reference angular position of cart S inside the circle	$[\textit{rad}]$
$\qquad\gamma_{S}^{*}(t)$	Reference angular position of the pan of the RGB-D sensor on cart S	$[\textit{rad}]$
$\qquad\alpha_{S}^{*}(t)$	Reference angular position of the tilt of the RGB-D sensor on cart S	$[\textit{rad}]$
$\mathbf{T_{Q}}(t)=[T_{1Q}(t),T_{2Q}(t),T_{3Q}(t)]^{T}$	Torque vector of the motors of cart Q and RGB-D sensor	$[N\cdot m]$
$\qquad T_{1Q}(t)$	Torque produced by the motors of cart Q	$[N\cdot m]$
$\qquad T_{2Q}(t)$	Torque produced by the pan motor of the RGB-D sensor of cart Q	$[N\cdot m]$
$\qquad T_{3Q}(t)$	Torque produced by the tilt motor of the RGB-D sensor of cart Q	$[N\cdot m]$
$\mathbf{T_{S}}(t)=[T_{1S}(t),T_{2S}(t),T_{3S}(t)]^{T}$	Torque vector of the motors of cart S and RGB-D sensor	$[N\cdot m]$
$\qquad T_{1S}(t)$	Torque produced by the motors of cart S	$[N\cdot m]$
$\qquad T_{2S}(t)$	Torque produced by the pan motor of the RGB-D sensor of cart S	$[N\cdot m]$
$\qquad T_{3S}(t)$	Torque produced by the tilt motor of the RGB-D sensor of cart S	$[N\cdot m]$
$I_{gp},I_{gt}$	Inertial torques of the camera’s rotational bodies	$[kg\cdot m^{2}]$
$I_{g}$	Inertial torque of the sprocket	$[kg\cdot m^{2}]$
$R_{r}$	Radius of the sprocket	$[m]$
$R$	Radius of the rail	$[m]$
$z_{c}$	Constant value of the RGB-D sensor height	$[m]$

Figure 1.

Proposed circular motorised rail system: (a) Initial configuration and; (b) Final configuration.

Figure 2 illustrates the general scheme of the platform, which consists of the following components:

(i) 1.

A motorised cart system composed of two carts mounted on the circular rail and equipped with an RGB-D sensor. The aim of this system is to relocate the two motorised carts to monitor the patient’s face and the physical rehabilitation exercise from two complementary viewpoints.

Two motorised carts equipped with an RGB-D device each (see Fig. 3), which move on the rail in a continuous manner to register the rehabilitation exercises and the facial expressions of the patient who freely moves inside the circle formed by the rail.

Figure 2.

General scheme of the platform.

An intelligent trajectory generator that generates the coordinate trajectories of the two carts to locate them in the optimal configuration to monitor the patient. The intelligent trajectory generator comprises the references required to move the carts (denoted as $\mathbf{x_{Q}^{*}}(t)$ and $\mathbf{x_{S}^{*}}(t)$ ) based on the instantaneous position of the patient in the circular rail (denoted as $\mathbf{P}(t)$ ), the instantaneous position of the part of the body to be monitored in the circular rail (denoted as $\mathbf{P_{b}}(t)$ ), the direction the person is looking at (denoted as principal vector, $\mathbf{v_{p}}(t)$ ), the direction to which the exercised part of body is pointing (denoted as secondary vector, $\mathbf{v_{s}}(t)$ ), and the instantaneous positions of the two carts (denoted as $\mathbf{x_{Q}}(t)$ and $\mathbf{x_{S}}(t)$ ). In order to perform the reference trajectories required to move the carts, it is necessary to establish the role (master or slave) of each cart, the direction (clockwise or counter-clockwise) that each cart must follow, and to generate the trajectories that avoid collisions between both carts.

Figure 3.

Layout of the cart equipped with an RGB-D sensor.

A GPI control action for each cart that uses the reference trajectories constructed by the trajectory generator, generating the necessary control actions in order to achieve both the stabilisation of the system and the tracking of the references (i.e. $\mathbf{x_{Q}}(t)\rightarrow\mathbf{x_{Q}^{*}}(t)$ and $\mathbf{x_{S}}(t)\rightarrow\mathbf{x_{S}^{*}}(t)$ ) and, consequently, to correctly monitor the patient (see Fig. 2).

3. Dynamic model of the carts

The dynamic model has been developed using as a basis the cart system shown in Fig. 3. The cart incorporates smooth support wheels, which are mobile supports that enable the cart to move, along with a rack and pinion that transmits its movement. The cart has been divided into three subsystems, which will make it possible to compute the balance of forces. These subsystems are the RGB-D sensor, the body of the cart and the sprocket. The calculations first consider the RGB-D sensor, after which they follow the transmission of forces to the motor that moves the sprocket. The dynamic model is obtained by attaining the function responsible for providing the torque generated by the motors. This will be the input to the system and is, therefore, important because from here on the system equations composing our state model are obtained.

The following expressions are calculated for the motors’ torques:

$\displaystyle T_{1}(t)=\left(\frac{I_{g}}{R}+mR_{r}\right)\cdot\ddot{x}(t)$ $\displaystyle T_{2}(t)=I_{gp}\cdot\ddot{\gamma}(t)$ (1) $\displaystyle T_{3}(t)=I_{gt}\cdot\ddot{\alpha}(t)$

where $I_{gp}$ and $I_{gt}$ denote the inertial torques of the camera’s rotational bodies, $I_{g}$ represents the inertial torque of the sprocket, $T_{1}(t)$ is the torque produced by the cart’s motors, $T_{2}(t)$ denotes the torque produced by the pan motor of the RGB-D sensor, $T_{3}(t)$ expresses the torque produced by the tilt motor of the RGB-D sensor, $m$ is the mass of the whole system, $R_{r}$ and $R$ express the radius of the sprocket and the rail, respectively, $x(t)$ represents the linear position of the track, $\gamma(t)$ is the angular position of the RGB-D sensor’s pan and $\alpha(t)$ represents the angular position of the RGB-D sensor’s tilt. Bearing in mind that the cart moves on a circular rail with a radius $R$ , it is necessary to transform the linear acceleration component for the motor of the cart, $\ddot{x}(t)$ , into an angular acceleration, defined as $\ddot{\beta}(t)$ , around the circle that the rail forms. The dynamic model is, therefore, the following:

$\displaystyle\ddot{\beta}(t)=\frac{R_{r}R}{I_{g}+mR^{2}_{r}}\cdot T_{1}(t)$ $\displaystyle\ddot{\gamma}(t)=\frac{1}{I_{gp}}\cdot T_{2}(t)$ (2) $\displaystyle\ddot{\alpha}(t)=\frac{1}{I_{gt}}\cdot T_{3}(t)$

or, written in a compact form:

$\displaystyle\underbrace{\left[\begin{array}[]{c}\ddot{\beta}(t)\\ \ddot{\gamma}(t)\\ \ddot{\alpha}(t)\\ \end{array}\right]}_{\mathbf{\ddot{x}}(t)}=\underbrace{\left[\begin{array}[]{% ccc}\varphi_{\beta}&0&0\\ 0&\varphi_{\gamma}&0\\ 0&0&\varphi_{\alpha}\end{array}\right]}_{\mathbf{N}}\cdot\underbrace{\left[% \begin{array}[]{c}T_{1}(t)\\ T_{2}(t)\\ T_{3}(t)\\ \end{array}\right]}_{\mathbf{T}(t)}$ (3)

where $\mathbf{N}=\textit{diag}(\varphi_{\beta},\varphi_{\gamma},\varphi_{\alpha})$ $\in\mathbb{R}^{3\times 3}$ is a diagonal positive definite matrix, being $\varphi_{\beta}=\frac{R_{r}R}{I_{g}+mR^{2}_{r}}$ , $\varphi_{\gamma}=\frac{1}{I_{gp}}$ and $\varphi_{\alpha}=\frac{1}{I_{gt}}$ . Figure 4 illustrates a visual description of the aforementioned variables. Finally, we should state that the dynamic model is used for both carts and is denoted with sub-indexes $(\cdot)_{Q}$ and $(\cdot)_{S}$ , respectively.

Figure 4.

Variables’ definition of a cart equipped with an RGB-D sensor.

4. Smart trajectory generator

The proposed trajectory generator must calculate the displacements of both carts inside the circle formed by the rail and the angular positions of the pan and tilt angles of the RGB-D sensors (i.e. $\mathbf{x_{Q}^{*}}(t)$ and $\mathbf{x_{S}^{*}}(t)$ ) in order to locate them in the optimal configuration to track the patient’s face and the physical rehabilitation exercises. For this reason, one of the carts is dynamically assigned the role of master and is responsible for following the patient’s face, while the other cart is assigned the role of slave and is responsible for monitoring the part of the patient’s body that is performing a particular rehabilitation exercise. To carry this out, the following information is required about both the patient and the carts (see Fig. 4):

•
the instantaneous position of the patient in the circular rail (denoted as $\mathbf{P}(t)\!=\![x_{p}(t),y_{p}(t),z_{p}(t)]^{T}$ with origin in point $O$ ),
•
the instantaneous position of the body part to be monitored in the circular rail (denoted as $\mathbf{P_{b}}(t)=[x_{b}(t),y_{b}(t),z_{b}(t)]^{T}$ with origin in point $O$ ),
•
the direction towards which the patient is facing (denoted as principal vector, $\mathbf{v_{p}}(t)=[x_{v}(t)$ , $y_{v}(t),z_{v}(t)]^{T}$ with origin located in the point $\mathbf{P}(t)$ ),
•
the direction to which the patient’s body part to be monitored is pointing (denoted as secondary vector, $\mathbf{v_{s}}(t)=[x_{s}(t),y_{s}(t),z_{s}(t)]^{T}$ with origin located in the point $\mathbf{P_{b}}(t)$ ), and,
•
the instantaneous configurations of the two carts (denoted as $\mathbf{x_{Q}}(t)=[\beta_{Q}(t),\gamma_{Q}(t),\alpha_{Q}(t)]^{T}$ and $\mathbf{x_{S}}(t)=[\beta_{S}(t),\gamma_{S}(t),\alpha_{S}(t)]^{T}$ ) which are inside the circle formed by the rail.

Based on this information, the intelligent trajectory generator must perform the following functions:

(i)
generate the optimal configuration of both carts, $\mathbf{x_{Q}^{}}(t)=[\beta^{}_{Q}(t),\gamma^{}_{Q}(t),\alpha^{}_{Q}(t)]^{T}$ and $\mathbf{x_{S}^{}}(t)=[\beta^{}_{S}(t),\gamma^{}_{S}(t),\alpha^{}_{S}(t)]^{T}$ , where $\beta^{}_{Q}(t)$ and $\beta^{}_{S}(t)$ represent the reference angular positions of carts $Q$ and $S$ around the circles, $\gamma^{}_{Q}(t)$ and $\gamma^{}_{S}(t)$ are the reference angular positions of the RGB-D sensors’ pan mounted on carts $Q$ and $S$ , and $\alpha^{}_{Q}(t)$ and $\alpha^{}_{S}(t)$ express the reference angular positions of the RGB-D sensors’ tilt mounted on carts $Q$ and $S$ ;
(ii)
establish the role (master or slave) of each cart in such manner that the master cart will be in charge to track the patient’s face and the slave cart will be assigned to monitor physical rehabilitation exercises;
(iii)
determine the direction that each cart must follow in order to minimise the distance travelled;
(iv)
avoid the possible collisions between both carts.

4.1 Angular positions of the carts, and pan and tilt angles of the cameras

The development of the mathematical model that is carried out for the computation of the reference trajectories of the carts when they act with the master or slave roles, denoted as

$\displaystyle\mathbf{x_{\textit{MS}}^{*}}(t)=[\beta^{*}_{\textit{MS}}(t),% \gamma^{*}_{\textit{MS}}(t),\alpha^{*}_{\textit{MS}}(t)]^{T}\text{and}$ $\displaystyle\mathbf{x_{\textit{SL}}^{*}}(t)=[\beta^{*}_{\textit{SL}}(t),% \gamma^{*}_{\textit{SL}}(t),\alpha^{*}_{\textit{MS}}(t)]^{T},$

respectively, are introduced. Let us highlight that the behaviours of the carts are different depending on the master and slave roles assigned. These roles are dynamically established according to the instantaneous locations of both carts. Moreover, the angular position of the slave cart must vary when a collision is foreseen with the master cart after the trajectories have been calculated. In this case, the slave cart must follow the opposite path. We have to remark that we will focus our study in the development of the mathematical procedure for obtaining the reference trajectories of the master cart, $\mathbf{x_{\textit{MS}}^{*}}(t)$ . According to Fig. 4, the computation of these reference trajectories uses the instantaneous position of the patient in the circular rail, $\mathbf{P}(t)$ , the direction the person is looking at, $\mathbf{v_{p}}(t)$ , and the instantaneous configuration of the master cart, $\mathbf{x}(t)=[\beta(t),\gamma(t),\alpha(t)]$ .

The procedure for achieving the reference trajectories for the slave cart, $\mathbf{x_{SL}^{*}}(t)$ , is similar to the model developed for the master cart, but using the instantaneous position of the part of the body to be monitored in the circular rail, $\mathbf{P_{b}}(t)$ , instead of the instantaneous position of the patient in the circular rail, $\mathbf{P}(t)$ , and using the direction to which the monitored part of the patient’s body is pointing, $\mathbf{v_{s}}(t)$ , instead of the direction in which the patient is facing, $\mathbf{v_{p}}(t)$ . For that reason, the procedure for obtaining $\mathbf{x_{SL}^{*}}(t)$ has not been included.

We start the development of the model by obtaining the new position of the master cart in the circular rail, $\beta^{*}_{MS}(t)$ , from which to observe the direction the person is looking at, is given by one of the two points, denoted as $\mathbf{P_{1}}=[x_{1},y_{1}]^{T}$ and $\mathbf{P_{2}}=[x_{2},y_{2}]^{T}$ , of the circular rail that intersects with the imaginary line $\mathbf{R_{p}}$ that results from the projection of the direction the patient is looking at, $\mathbf{v_{p}}(t)$ , on the $X Y$ -plane and whose origin is in his/her current position $\mathbf{P}(t)$ . Next, we will study which of the two points focuses on the said vector, discarding the other and, finally, we will obtain the position to which the cart will move. This approach allows to geometrically obtain the two cut points of the line passing through the patient’s position, taking the patient’s face as the direction. Figure 5 illustrates a visual description of the angular positions obtained for the master cart.

Figure 5.

Angular positions of the master cart.

In order to obtain the mathematical description of the line $\mathbf{R_{p}}$ , the patient’s position, $\mathbf{P}(t)$ , and the direction the patient is looking at, $\mathbf{v_{p}}(t)$ , are used, obtaining:

$\displaystyle\frac{x-x_{p}}{x_{v}}=\frac{y-y_{p}}{y_{v}}$ (4)

where $x$ and $y$ are the coordinates of the different points that belong to line $\mathbf{R_{p}}$ . The parametrisation of the circular rail of radius $R$ is defined as:

$\displaystyle x^{2}+y^{2}=R^{2}$ (5)

Upon operating with Eqs (4) and (5), we obtain:

$\displaystyle x=\frac{x_{v}}{y_{v}}\cdot y+\left(x_{p}-\frac{y_{p}}{y_{v}}% \cdot x_{v}\right)$ (6) $\displaystyle\left(\frac{x_{v}}{y_{v}}\cdot y+\left(x_{p}-\frac{y_{p}}{y_{v}}% \cdot x_{v}\right)\right)^{2}+y^{2}=R^{2}$ (7)

By defining $m=\frac{x_{v}}{y_{v}}$ and $n=x_{p}-\frac{y_{p}}{y_{v}}\cdot x_{v}$ , Eq. (7) can be expressed in the following compact form:

$\displaystyle(my+n)^{2}+y^{2}=R^{2}$ (8)

Now, after simple algebraic manipulations in Eq. (8), the following result is obtained:

$\displaystyle(m^{2}+1)y^{2}+2mny+(n^{2}-R^{2})=0$ (9)

The values of the resulting cut points $\mathbf{P_{1}}=[x_{1},y_{1}]^{T}$ and $\mathbf{P_{2}}=[x_{2},y_{2}]^{T}$ , as depicted in Fig. 5, are computed as follows:

$\displaystyle y_{1}=\frac{-2mn+\sqrt{4m^{2}n^{2}-4(m^{2}+1)(n^{2}-R^{2})}}{2(m% ^{2}+1)}$ (10) $\displaystyle x_{1}=(y_{1}-y_{p})\cdot m+x_{p}$ $\displaystyle y_{2}=\frac{-2mn-\sqrt{4m^{2}n^{2}-4(m^{2}+1)(n^{2}-R^{2})}}{2(m% ^{2}+1)}$ (11) $\displaystyle x_{2}=(y_{2}-y_{p})\cdot m+x_{p}$

In order to discover which point is correct, the parametric equation of line $\mathbf{R_{p}}$ with the value of parameter $h=\frac{x_{1}-x_{p}}{x_{v}}=\frac{y_{1}-y_{p}}{y_{v}}$ is calculated by substituting the value of the first gotten point in said equation. If $h\geqslant 0$ , then point $\mathbf{P_{1}}$ is the cart’s destination point. If $h<0$ , then the cart must move to point $\mathbf{P_{2}}$ . The next step consists in using parameter $\beta_{M}(t)=\arctan{\left(\frac{y_{1}}{x_{1}}\right)}$ to convert that point of the circumference into an angular position around the circle. Once we have obtained the point to be addressed, the movement of the cart has two possible solutions. The first solution is $\Delta\beta_{1}(t)=\beta_{M}(t)-\beta(t)$ , while the second corresponds to the opposite direction. If $\Delta\beta_{1}(t)\geqslant 0$ , which corresponds to a clockwise turn, we have $\Delta\beta_{2}(t)=2\pi+\beta_{M}(t)-\beta(t)$ . If the turn is counter-clockwise (i.e. $\Delta\beta_{1}(t)<0$ ), we have $\Delta\beta_{2}(t)=-2\pi+\beta_{M}(t)-\beta(t)$ . Bearing in mind all the aforementioned, the new position of the master cart in the circular rail, $\beta^{*}_{MS}(t)$ , is:

$\displaystyle\beta^{*}_{\textit{MS}}(t)=\min(|\Delta\beta_{1}(t)|,|\Delta\beta% _{2}(t)|)$ (12)

The following step is to calculate the angular position of the RGB-D sensor’s pan of the master cart, $\gamma^{*}_{\textit{MS}}(t)$ , which is calculated as follows. The direction of the pan vector of the RGB-D sensor of the master cart, whose angle is $\gamma(t)$ , moves until it occupies the opposite direction to the principal vector of the patient, i.e., $\gamma_{p}(t)+\pi$ , where $\gamma_{p}(t)$ is defined as follows (see Fig. 5):

$\displaystyle\gamma_{p}(t)=\arctan{\left(\frac{y_{v}(t)-y_{p}(t)}{x_{v}(t)-x_{% p}(t)}\right)}$ (13)

However, as the RGB-D sensor is mounted on a cart rotating around the rail, the turn $\beta^{*}_{\textit{MS}}(t)$ made by the cart needs to be subtracted, thus obtaining:

$\displaystyle\Delta\gamma_{1}(t)=\gamma(t)-(\gamma_{p}(t)+\pi)-\beta^{*}_{% \textit{MS}}(t)$ (14) $\displaystyle\Delta\gamma_{2}(t)=\pi-\gamma(t)+\gamma_{p}(t)-\beta^{*}_{% \textit{MS}}(t)$ (15)

where $\Delta\gamma_{1}(t)$ and $\Delta\gamma_{2}(t)$ represent the two possible turn solutions of the pan movement of the RGB-D sensor based on the turn of the cart (clockwise or counter-clockwise) around the rail. After computing Eqs (14) and (15), the angular position of the RGB-D sensor’s pan of the master cart, $\gamma^{*}_{\textit{MS}}(t)$ , is the following:

$\displaystyle\gamma^{*}_{\textit{MS}}(t)=\min(|\Delta\gamma_{1}(t)|,|\Delta% \gamma_{2}(t)|)$ (16)

Figure 6.

Angular position of the the master cart’s tilt sensor.

Next, the angular position of the tilt of the RGB-D sensor of the master cart, $\alpha^{*}_{\textit{MS}}(t)$ , is calculated. In this case, the direction of the tilt vector of the RGB-D sensor of the master cart, whose angle is $\alpha(t)$ , moves until it occupies the opposite direction to the principal vector of the patient in the vertical plane defined by angle $\gamma_{p}(t)$ (see Fig. 5), i.e., $\alpha_{p}(t)+\pi$ , where $\alpha_{p}(t)$ is defined as follows:

$\displaystyle\alpha_{p}(t)\!=\!\arctan{\left(\!\!\!\right)}$ (17)

where $z_{c}$ denotes the constant value of the RGB-D sensor height. Moreover, the movement of the RGB-D sensor is obtained by considering that the cart is also moving. The objective is, therefore, for $\alpha^{*}_{\textit{MS}}(t)$ to occupy the place of the angle opposite to $\alpha_{p}(t)$ , that is, $\alpha_{p}(t)+\pi$ . There are again two rotation possibilities to ensure that the RGB-D sensor reaches the desired point:

$\displaystyle\Delta\alpha_{1}(t)=\alpha_{p}(t)-(\pi+\alpha(t))$ (18) $\displaystyle\Delta\alpha_{2}(t)=\pi-\alpha_{p}(t)+\alpha(t)$ (19)

Again, after computing Eqs (18) and (19), the angular position of the tilt of the RGB-D sensor of the master cart, $\alpha^{*}_{MS}(t)$ , is the following:

$\displaystyle\alpha^{*}_{\textit{MS}}(t)=\min(|\Delta\alpha_{1}(t)|,|\Delta% \alpha_{2}(t)|)$ (20)

Finally, grouping Eqs (12), (16) and (20), the final reference trajectory vector for the master cart, $\mathbf{x_{\textit{MS}}^{*}}(t)$ , is shown next:

$\displaystyle\mathbf{x_{\textit{MS}}^{*}}(t)=\left[\begin{array}[]{c}\beta^{*}% _{\textit{MS}}(t)\\ \gamma^{*}_{\textit{MS}}(t)\\ \alpha^{*}_{\textit{MS}}(t)\\ \end{array}\right]=\left[\begin{array}[]{c}\min(|\Delta\beta_{1}(t)|,|\Delta% \beta_{2}(t)|)\\ \min(|\Delta\gamma_{1}(t)|,|\Delta\gamma_{2}(t)|)\\ \min(|\Delta\alpha_{1}(t)|,|\Delta\alpha_{2}(t)|)\\ \end{array}\right]$ (21)

4.2 The carts’ roles

The selection of each cart’s role is determined according to the following criteria:

1.
When the destination angular positions of the cart that are required to cover the patient’s position have been calculated, the cart closest to the main vector, defined by the angular position $\beta_{M}(t)$ , is the first to become the master. The other cart is assigned the role of slave. However, this may change, as the total route followed by both carts is priority when choosing the roles.
2.
Any collision between the carts must be avoided, and it is consequently necessary to check that the route used by one cart is not interspersed with the other, thus avoiding such a situation.
3.
If both carts are at the same distance from angular position $\beta_{M}(t)$ , the cart closest to angular position $\beta_{E}(t)$ becomes the slave.
4.
If both angular positions are at the same distance from the two carts, the one whose speed in the direction of the angular position is greater takes the role of master.
5.
If both carts are travelling at the same speed or are stationary, cart $Q$ will be the master and cart $S$ will be the slave, i.e. $\mathbf{x_{Q}^{}}(t)=\mathbf{x_{\textit{MS}}^{}}(t)$ and $\mathbf{x_{S}^{}}(t)=\mathbf{x_{\textit{SL}}^{}}(t)$ .

In order to explain these situations, two examples are analysed (see Fig. 7), where it is possible to determine which of the two carts should go to each position according to the destination.

Figure 7.
Examples of role distribution.

In Fig. 7a, the distribution of roles is very simple according to the first rule described. Cart $Q$ takes the role of master, $\mathbf{x_{Q}^{}}(t)=\mathbf{x_{\textit{MS}}^{}}(t)$ , and moves to position $\beta_{M}(t)$ , since it is the closest, while cart $S$ plays the role of slave, $\mathbf{x_{S}^{}}(t)=\mathbf{x_{SL}^{}}(t)$ , and moves to position $\beta_{E}(t)$ . In Fig. 7b, the same assignment of roles occurs. Cart $Q$ would be the master and go to point $\beta_{M}(t)$ , while cart $S$ would take on the role of slave and go to point $\beta_{E}(t)$ . However, in this case, the trajectory planner foresees a collision between both carts for the shortest path (both carts moving in a counter-clockwise manner). Two possible options appear for the movement of the carts:

(i)
cart $Q$ acts as the master and moves to point $\beta_{M}(t)$ counter-clockwise and cart $S$ acts as the slave and moves to point $\beta_{E}$ clockwise;
(ii)
a change of roles of the carts is necessary and cart $Q$ takes on the role of slave and cart $S$ becomes the slave and both carts move in a counter-clockwise manner.

The intelligent trajectory planner studies these options and selects the optimal movement of the carts (which is given by the minimum path travelled by the two carts) and immediately changes the roles of the carts, with cart $Q$ playing the role of slave and $S$ taking on the role of master, i.e. $\mathbf{x_{Q}^{}}(t)=\mathbf{x_{\textit{SL}}^{}}(t)$ and $\mathbf{x_{S}^{}}(t)=\mathbf{x_{\textit{MS}}^{}}(t)$ .

We conclude by stating the main results regarding the design of the intelligent trajectory planner:

Given the instantaneous position of the patient in the circular rail, $\mathbf{P}(t)=[x_{p}(t),y_{p}(t),z_{p}(t)]^{T}$ , the instantaneous position of the part of the body to be monitored in the circular rail, $\mathbf{P_{b}}(t)=[x_{b}(t),y_{b}(t),z_{b}(t)]^{T}$ , the direction to which the patient is facing, $\mathbf{v_{p}}(t)=[x_{v}(t),y_{v}(t),z_{v}(t)]^{T}$ , the direction to which the part of the patient’s body to be monitored is pointing, $\mathbf{v_{s}}(t)=[x_{s}(t),y_{s}(t)$ , $z_{s}(t)]^{T}$ , and the instantaneous configuration of the two carts which are inside the circle formed by the rail, $\mathbf{x_{Q}}(t)=[\beta_{Q}(t),\gamma_{Q}(t),\alpha_{Q}]^{T}$ and $\mathbf{x_{S}}(t)=[\beta_{S}(t),\gamma_{S}(t),\alpha_{S}]^{T}$ , the intelligent trajectory planner defined by Eqs (4)–(21) and the criteria provided in Section 4.2 for the selection of the carts’ roles applied to the carts, establishes the role of each cart (master or slave), determines the direction (clockwise or counter-clockwise) that each cart must follow, and generates the trajectories that will avoid collisions between the carts in such a manner that the displacement of both carts (i.e. $\mathbf{x_{Q}^{}}(t)=[\beta^{}_{Q}(t),\gamma^{}_{Q}(t),\alpha^{}_{Q}(t)]^{T}$ and $\mathbf{x_{S}^{}}(t)=[\beta^{}_{S}(t),\gamma^{}_{S}(t),\alpha^{}_{S}(t)]^{T}$ ) are determined in order to locate them in the optimal configuration to track the patient’s face and the physical rehabilitation exercises.

5. GPI Control for the carts

In recent years, novel robust control algorithms have been developed for smart mechanical structures [32, 33, 34, 35]. In the case of the circular rail, the generalised proportional integral (GPI) control, which has performed well in regulating the dynamic systems of a linear and non-linear nature [36, 37], has been applied to control the movement of the two carts in an intelligent and precise manner. The main advantage of GPI control is that it sidesteps the need for traditional asymptotic state observers and directly proceeds to use, in a previously designed state feedback control law, defective integral reconstruction of the state. This, a priori, neglects the effects of unknown initial conditions and possible class-ical perturbation inputs (constant and low-order time polynomial errors, such as ramps and parabolic signals).

These structural estimates are based on the key observation that the states of observable linear systems may be integrally parametrised in terms of inputs and outputs alone (i.e. linear combinations of inputs, outputs and of a finite number of iterated integrals of signals). The errors of integral reconstruction are compensated later by means of a sufficiently large number of additional iterated integral tracking errors, integral input errors and control actions (see [38] for the relevant theoretical basis and [39, 40, 41, 42, 43, 44, 45] for the application of these ideas in diverse fields, including laboratory experiments).

In this work, we extend the GPI control technique to both the stabilisation and trajectory tracking tasks of the two carts that comprise the circular motorised platform. We first derive an exact controller to be used in the control of the configuration of the carts, under the assumption that there is complete knowledge of the system states of the two carts. The velocity vector estimation of the carts is then carried out by means of a model-based integral re-constructor complemented with a GPI controller. Finally, we prove that the proposed GPI controller scheme for each of the carts gene-rates tracking errors tending to zero, thus robustly tracking the patient’s face and/or his/her physical rehabilitation exercises.

5.1 Robust feed-forward controller for the motorised circular platform

Firstly, it is first necessary to recall that both carts have the same mechanical components, and thus have the same dynamic model. In this case, the derivation of the controller can be performed in a general manner, after which the sub-indexes $(\cdot)_{Q}$ and $(\cdot)_{S}$ are respectively included to obtain the controller of the two carts on the platform. Now, considering the dynamics of the cart given by expression (3), where it is aimed to track a desired configuration reference trajectory vector for the cart denoted as $\mathbf{x^{*}}(t)$ . Clearly, if there exists an open loop control input vector, $\mathbf{T}^{*}(t)$ , that ideally obtains the tracking of the desired configuration reference trajectory vector for cart $\mathbf{x^{*}}(t)$ through appro-priate initial conditions, it satisfies the following expression:

$\displaystyle\mathbf{\ddot{x}^{*}}(t)=\mathbf{N}\cdot\mathbf{T^{*}}(t)$ (22)

If Eq. (22) is subtracted to Eq. (3), the tracking error vector $\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x^{*}}(t)$ satisfies the following second order dynamics:

$\displaystyle\mathbf{\mathbf{\ddot{e}}}(t)=\mathbf{N}\cdot(\mathbf{T}(t)-% \mathbf{T^{*}}(t))$ (23)

A direct exact linearisation-based feedback controller that exponentially regulates to zero the tracking error vector $\mathbf{\mathbf{e}}(t)$ is given by the following expression:

$\displaystyle\mathbf{T}(t)=\mathbf{T^{*}}(t)+\mathbf{N}^{-1}\cdot\bm{\nu}(t)$ (24)

where

$\displaystyle\bm{\nu}(t)=-\mathbf{K_{2}}\mathbf{\dot{e}}(t)-\mathbf{K_{1}}% \mathbf{e}(t)=-\mathbf{K_{2}}\left(\mathbf{\dot{x}}(t)-\mathbf{\dot{x}^{*}}(t)% \right)\!-\!\mathbf{K_{1}}\left(\mathbf{x}(t)-\mathbf{x^{*}}(t)\right)$ (25)

where $\mathbf{K_{2}}$ and $\mathbf{K_{1}}$ $\in\mathbb{R}^{3\times 3}$ are diagonal positive definite matrices that represent the design elements of a vector-valued controller. Upon the appropriate choice of the design vector matrices $\mathbf{K_{2}}$ and $\mathbf{K_{1}}$ , the preceding controller imposes the following exponentially asymptotically stable closed-loop dynamics on the tracking error vector $\mathbf{e}(t)$ :

$\displaystyle\mathbf{\ddot{e}}(t)+\mathbf{K_{2}}\mathbf{\dot{e}}(t)+\mathbf{K_% {1}}\mathbf{e}(t)=\mathbf{0}$ (26)

We shall conclude this section by stating the following result, which has been proved in the developments shown above:

Given the dynamics of each of the carts Eq. (3), of which the circular motorised platform illustrated in Fig. 2 is composed, where it is assumed that the velocity vector system, $\mathbf{\dot{x}}(t)=[\dot{\beta}(t),\dot{\gamma}(t),\dot{\alpha}(t)]^{T}$ is perfectly measurable, and given a smooth reference trajectory vector $\mathbf{x^{*}}(t)$ for the configuration of the cart in the circular platform provided by the intelligent trajectory generator, then, with the appropriate choice of the design vector matrices $\mathbf{K_{2}}$ and $\mathbf{K_{1}}$ $\in\mathbb{R}^{3\times 3}$ such that the closed loop cha-racteristic 3 $\times$ 3 complex valued diagonal matrices $\mathbf{I}^{3\times 3}s^{2}+\mathbf{K_{2}}s+\mathbf{K_{1}}=\mathbf{0}$ are all second-degree Hurwitz polynomials, the control law given by expressions (24)–(25) provides the desired tracking error equilibrium vector $\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x^{*}}(t)=\mathbf{0}$ , which is of an exponentially asymptotically stable nature.

An alternative complementary design that requires no perfect knowledge of the velocity vector system, $\mathbf{\dot{x}}(t)=[\dot{\beta}(t),\dot{\gamma}(t),\dot{\alpha}(t)]^{T}$ is described next.

5.2 Integral re-constructor for the velocity vector system, and GPI controller design for the circular motorised platform

The control law given by Eqs (24) and (25) requires the perfect knowledge of the velocity vector system, $\mathbf{\dot{x}}(t)=[\dot{\beta}(t),\dot{\gamma}(t),\dot{\alpha}(t)]^{T}$ . This vector is customarily quite noisy and the use of low-pass filters is usually necessary to smoothing it. This causes the well-known dynamic delays that affect the performance and accuracy of the tracking system [46, 47, 48, 49]. During the development of a GPI controller, the computation of the estimate velocity vector, $\mathbf{\dot{\hat{x}}}(t)$ , requires only a linear combination of iterated integrals of the input vector, $\mathbf{T}(t)$ , and the output vector of the system, $\mathbf{x}(t)$ , thus avoiding the aforementioned problems. Indeed, we suppose that only the configuration vector of the cart $\mathbf{x}(t)$ and the input vector $\mathbf{T}(t)$ are available for measurement. In this case, the velocity vector system $\mathbf{\dot{x}}(t)$ is directly determined by integration of Eq. (3) as follows:

$\displaystyle\mathbf{\dot{x}}(t)=\mathbf{N}\cdot\int_{0}^{t}{\mathbf{T}(\sigma% )d\sigma}+\mathbf{\dot{x}}(0)$ (27)

where $\mathbf{\dot{x}}(0)$ expresses the value of the velocity vector, at time $t=0$ . However, this information is usually not available, and we can only assume that this quantity is known in the simplest cases (i.e. when it is known that the motorised cart starts with a null velocity vector). We, therefore, have no knowledge of the value of magnitude $\mathbf{\dot{x}}(0)$ . Let us suppose for a moment that, motivated by the simplicity of Eq. (27), we insist upon using the following structural estimation for the velocity vector of the motorised cart $\mathbf{\dot{x}}(t)$ :

$\displaystyle\mathbf{\dot{\hat{x}}}(t)=\mathbf{N}\cdot\int_{0}^{t}{\mathbf{T}(% \sigma)d\sigma}$ (28)

The structural velocity vector estimator provided in Eq. (28) certainly has the interesting advantage of being easily synthesised by using only the integral of a linear function of input vector $\mathbf{T}(t)$ . The exact relationship between the structural velocity vector estimation and the actual value of the velocity vector of the motorised cart is evidently obtained after inspection of Eqss (27) and (28) from the following expression:

$\displaystyle\mathbf{\dot{x}}(t)=\mathbf{\dot{\hat{x}}}(t)+\mathbf{\dot{x}}(0)$ (29)

The following equivalent feedback control law is, therefore, obtained by substituting Eqs (28) and (29) in controller Eqs (24) and (25):

$\displaystyle\mathbf{\tilde{T}}(t)=\mathbf{T^{*}}(t)+\mathbf{N}^{-1}\cdot\bm{% \tilde{\nu}}(t)$ $\displaystyle\bm{\tilde{\nu}}(t)=-\mathbf{K_{2}}\!\left(\mathbf{\dot{\hat{x}}}% (t)-\mathbf{\dot{x}^{*}}(t)\!\right)\!-\!\mathbf{K_{1}}\left(\mathbf{x}(t)-% \mathbf{x^{*}}(t)\!\right)=-\mathbf{K_{2}}\left(\mathbf{N}\cdot\int_{0}^{t}{% \mathbf{T}(\sigma)d\sigma}-\mathbf{\dot{x}^{*}}(t)\right)-\mathbf{K_{1}}\left(% \mathbf{x}(t)-\mathbf{x^{*}}(t)\right)=-\mathbf{K_{2}}\left(\mathbf{\dot{x}}(t% )-\mathbf{\dot{x}}(0)-\mathbf{\dot{x}^{*}}(t)\right)-\mathbf{K_{1}}\left(% \mathbf{x}(t)-\mathbf{x^{*}}(t)\right)$ (30)

When the control given by Eq. (5.2) is implemented in the motorised cart model Eq. (3), we obtain that the tracking error dynamics evolves according to the following linear perturbed dynamics:

$\displaystyle\mathbf{\ddot{e}}(t)+\mathbf{K_{2}}\mathbf{\dot{e}}(t)+\mathbf{K_% {1}}\mathbf{e}(t)=\mathbf{K_{2}}\mathbf{\dot{x}}(0)$ (31)

which has an offset error owing to the constant initial condition excitation of the asymptotically stable tracking error vector dynamics. This immediately prompts us to consider the possibility of using a modified linear control action including an integral error term that rejects these constant perturbation loads. We, therefore, propose the following control law, which is illustrated in Fig. 8:

$\displaystyle\mathbf{T}(t)=\mathbf{T^{*}}(t)+\mathbf{N}^{-1}\cdot\bm{\nu}(t)$ $\displaystyle\bm{\nu}(t)=-\mathbf{K_{2}}\!\left(\!\mathbf{\dot{\hat{x}}}(t)-% \mathbf{\dot{x}^{*}}(t)\right)\!-\!\mathbf{K_{1}}\left(\mathbf{x}(t)-\mathbf{x% ^{*}}(t)\!\right)-\mathbf{K_{0}}\cdot\int^{t}_{0}{\left(\mathbf{x}(\sigma)-% \mathbf{x^{*}}(\sigma)\right)d\sigma}=-\mathbf{K_{2}}\left(\underbrace{\mathbf% {N}\cdot\int_{0}^{t}{\mathbf{T}(\sigma)d\sigma}}_{\mathbf{\dot{x}}(t)-\mathbf{% \dot{x}}(0)}-\mathbf{\dot{x}^{*}}(t)\right)-\mathbf{K_{1}}\left(\mathbf{x}(t)-% \mathbf{x^{*}}(t)\right)-\mathbf{K_{0}}\cdot\int^{t}_{0}{\left(\mathbf{x}(% \sigma)-\mathbf{x^{*}}(\sigma)\right)d\sigma}$ (32)

Figure 8.

GPI control scheme.

The use of the modified controller Eq. (5.2) in the dynamics of the motorised cart model Eq. (3) gives the following dynamics for the tracking error:

$\displaystyle\mathbf{\ddot{e}}(t)+\mathbf{K_{2}}\mathbf{\dot{e}}(t)+\mathbf{K_% {1}}\mathbf{e}(t)+$ (33) $\displaystyle\quad\mathbf{K_{0}}\cdot\int_{0}^{t}{\mathbf{e}(\sigma)d\sigma}=% \mathbf{K_{2}}\mathbf{\dot{x}}(0)$

Now, by differentiating Eq. (5.2), the following third-order linear tracking error vector dynamics is achieved:

$\displaystyle\mathbf{e}^{(3)}(t)+\mathbf{K_{2}}\mathbf{\ddot{e}}(t)+\mathbf{K_% {1}}\mathbf{\dot{e}}(t)+\mathbf{K_{0}}\mathbf{e}(t)=\mathbf{0}$ (34)

which clearly has the origin ( $\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x^{*}}(t)=\mathbf{0}$ ) as an asymptotically, exponentially stable equilibrium point, independently of the values of the initial conditions of the system. The constant design parameters $\{\mathbf{K_{2}},\mathbf{K_{1}},\mathbf{K_{0}}\}\in\mathbb{R}^{3\times 3}$ are diagonal positive definite matrices which are designed in such a way that the dominant characteristic $3\times 3$ complex valued diagonal matrix:

$\displaystyle\mathbf{p}(s)=\mathbf{I}^{3\times 3}s^{3}+\mathbf{K_{2}}s^{2}+% \mathbf{K_{1}}s+\mathbf{K_{0}}=\mathbf{0}$ (35)

is composed of third-degree Hurwitz polynomials with desirable root locations. The constant gain matrices $\{\mathbf{K_{2}},\mathbf{K_{1}},\mathbf{K_{0}}\}$ were designed in order to locate the desired closed-loop poles at a common root of the real line, given by $\mathbf{I}^{3\times 3}s=-\mathbf{p_{c}}$ , where $\mathbf{p_{c}}$ is a diagonal matrix whose terms are clearly strictly positive. We use the following $3\times 3$ complex valued diagonal matrix:

$\displaystyle\mathbf{p_{d}}(s)=\left(\mathbf{I}^{3\times 3}s+\mathbf{p_{c}}% \right)^{3}=\mathbf{0}$ (36)

Upon identifying each term in Eq. (35) with those in Eq. (36), we directly obtain the values required for the set of gain matrices $\{\mathbf{K_{2}},\mathbf{K_{1}},\mathbf{K_{0}}\}$ . The following result is obtained:

$\displaystyle\mathbf{K_{2}}=3\mathbf{p_{c}};\ \ \mathbf{K_{1}}=3\mathbf{p^{2}_% {c}};\ \ \mathbf{K_{0}}=\mathbf{p^{3}_{c}}$ (37)

We conclude by stating the main result regarding the design of the control law, which has been proved throughout all of the above:

Given the reference trajectory vectors provided by the intelligent trajectory generator, $\mathbf{x_{Q}^{*}}(t)$ and $\mathbf{x_{S}^{*}}(t)$ (which are necessary to locate the carts in the optimal configuration inside the circular rail in order to track the patient’s face and physical rehabilitation exercises) for the configurations of the carts, defined by $\mathbf{x_{Q}}(t)$ and $\mathbf{x_{S}}(t)$ , in which both carts share the dynamic model:

$\displaystyle\mathbf{\ddot{x}}(t)=\mathbf{N}\cdot\mathbf{T}(t)$ (38)

and, under the assumption that only measurements of the configuration of the carts, $\mathbf{x_{Q}}(t)$ and $\mathbf{x_{S}}(t)$ , and of the applied control inputs, $\mathbf{T_{Q}}(t)$ and $\mathbf{T_{S}}(t)$ , are available, then the following feedback controller action applied to each cart:

$\displaystyle\mathbf{T}(t)=\mathbf{T^{*}}(t)+\mathbf{N}^{-1}\cdot\bm{\nu}(t)$ $\displaystyle\bm{\nu}(t)=-\mathbf{K_{2}}\left(\mathbf{\dot{\hat{x}}}(t)-% \mathbf{\dot{x}^{*}}(t)\right)\quad-\mathbf{K_{1}}\left(\mathbf{x}(t)-\mathbf{% x^{*}}(t)\right)-\mathbf{K_{0}}\int^{t}_{0}{\left(\mathbf{x}(\sigma)-\mathbf{x% ^{*}}(\sigma)\right)d\sigma}=-\mathbf{K_{2}}\left(\mathbf{N}\int_{0}^{t}{% \mathbf{T}(\sigma)d\sigma}-\mathbf{\dot{x}^{*}}(t)\right)-\mathbf{K_{1}}\left(% \mathbf{x}(t)-\mathbf{x^{*}}(t)\right)-\mathbf{K_{0}}\int^{t}_{0}{\left(% \mathbf{x}(\sigma)-\mathbf{x^{*}}(\sigma)\right)d\sigma}$ (39)

produces a closed-loop behaviour of the tracking error vector, defined as $\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x^{*}}(t)$ , which is governed by the linear tracking error dynamics

$\displaystyle\mathbf{e}^{(3)}(t)\!+\!\mathbf{K_{2}}\mathbf{\ddot{e}}(t)\!+\!% \mathbf{K_{1}}\mathbf{\dot{e}}(t)\!+\!\mathbf{K_{0}}\mathbf{e}(t)\!=\!\mathbf{0}$ (40)

whose design parameters, $\{\mathbf{K_{2}},\mathbf{K_{1}},\mathbf{K_{0}}\}\in\mathbb{R}^{3\times 3}$ , can be selected so as to ensure that the origin of the tracking error vector space, $\mathbf{e}(t)=\mathbf{x}(t)-\mathbf{x^{*}}(t)=\mathbf{0}$ , is of an exponential asymptotically stable nature.

6. Numerical simulations

Numerical simulations were carried out in order to evaluate the performance of the intelligent trajectory generation and the control of the carts. For the numerical simulations, we considered the following values for the parameters of the carts and the circular rail: $I_{gp}=$ 0.1360 [kg $\cdot$ m ${}^{2}$ ], $I_{gt}=$ 0.0002 [kg $\cdot$ m ${}^{2}$ ], $I_{g}=$ 0.089 [kg $\cdot$ m ${}^{2}$ ], $m=$ 40 [kg], $R_{r}=$ 0.0375 [m], $z_{c}=$ 1 [m] and $R=$ 2.2 [m]. The sampling time was set to 0.02 [s]. The desired Hurwitz 3 $\times$ 3 complex diagonal matrix for the GPI controllers of the carts was $\mathbf{p_{c}}=\textit{diag}(1,1,1)$ .

Two case studies were developed to demonstrate the effectiveness of the proposed intelligent trajectory planner, which coincides with the case studies explained in Section 4.2. Additionally, in the case studies two different simulations were developed in order to establish a comparison between the GPI control presented in this paper and a standard PID control [50], denoted in the graphs as PID and whose control scheme is depicted in Fig. 9.

Figure 9.

PID control scheme.

6.1 Case study A

This case study was previously explained in Section 4.2 (see Fig. 7a). The instantaneous position of the patient in the circular rail is $\mathbf{P}=$ [ $-$ 0.5, $-$ 0.5, 0.5] ${}^{T}$ [m], the instantaneous position of the part of the patient’s body to be monitored is $\mathbf{P_{b}}=$ [ $-$ 0.5, $-$ 0.5, 1.2] ${}^{T}$ [m], the direction that the patient is facing is $\mathbf{v_{p}}=[-{\textstyle\frac{\sqrt{2}}{2}},{\textstyle\frac{\sqrt{2}}{2}}% ,0]^{T}$ , the direction to which the part of the patient’s body to be monitored is pointing, $\mathbf{v_{s}}=[1,0,0]^{T}$ , and the initial angular positions of the two carts which are inside the circle of the rail are $\mathbf{x_{Q_{0}}}=[{\textstyle\frac{\pi}{2}},{\textstyle\frac{7\pi}{4}},0]^{T}$ $[\textit{rad}]$ and $\mathbf{x_{S_{0}}}=[{\textstyle\frac{\pi}{4}},{\textstyle\frac{5\pi}{4}},0]^{T}$ $[\textit{rad}]$ , respectively.

Figure 10.

Positioning graphs of the case study A: (a) starting configurations; (b) final configurations.

Figure 10a shows the configurations of the carts and the patient at the initial simulation instant. In this example, the trajectory generator selects the shortest path and assigns the role of master to cart $Q$ and that of slave to $S$ . In addition, we compare the values calculated by the algorithm and the final path to check whether the system has selected the shortest route. Table 2 contains the aforementioned data.

Table 2

Distance according to path

Data (rad)	Cart $Q$	Cart $S$
Path clockwise	$-$ 5.1706	$-$ 1.0147
Path counter-clockwise	1.1126	5.2685
Path selected	1.1126	$-$ 1.0147

As can be verified, the system selects the shortest path between the possible routes (clockwise or counter-clockwise). That is, the route in the counter-clockwise direction for cart $Q$ and the route in clockwise direction for cart $S$ , respectively.

Figure 10b illustrates the final configurations reach-ed by the carts Q and S. Notice that the carts reach the desired positions, just as the RGB-D sensor has successfully rotated to point directly towards the patient. We also checked that the system has correctly selected the roles such that the total distance travelled by both carts is minimal (see Table 3). Since cart $Q$ was selected as master and cart $S$ as slave, it is verified that the trajectory generator has correctly selected the shortest distance travelled by both carts.

Table 3

Distance according to role

Data (rad)	$Q$ master, $S$ slave	$Q$ slave, $S$ master
Distance $Q$	1.1126	$-$ 1.8980
Distance $S$	$-$ 1.0147	1.8980
Total distance	2.1273	3.7960

The next checked parameter was related with the angular positions of the pan of the RGB-D sensors of the carts. In this case, the angular position of cart $Q$ ’s pan has to move to point the patient’s face (principal vector) with an angular position whose value is $\gamma_{p}(t)+\pi$ , while the angular position of cart $S$ has to move to point the patient’s body (secondary vector) with an angular position whose value is $\gamma_{e}(t)+\pi$ , where $\gamma_{e}(t)$ is defined as follows:

$\displaystyle\gamma_{e}(t)=\arctan{\left(\frac{y_{s}(t)-y_{b}(t)}{x_{s}(t)-x_{% b}(t)}\right)}$ (41)

The data obtained for the these parameters are summed up in Table 4, showing that the system has correctly selected the route required to travel the shortest path for the angular positions of both carts’ pan. In this case, the routes are counter-clockwise and clockwise for the angular positions of the pan of the RGB-D sensors of carts $Q$ and $S$ , respectively. Figure 10b illu-strates that the angular positions of the RGB-D sensors’ pan reach the final desired positions.

Table 4

Pan according to path

Data (rad)	Cart $Q$	Cart $S$
Path clockwise	$-$ 1.1126	6.0539
Path counter-clockwise	5.1706	0.2293
Path selected	$-$ 1.1126	0.2293

After that, it was necessary to check the angular positions of the RGB-D sensors’ tilt of the carts. In this case, the angular position of the tilt of cart $Q$ has to move to point the patient’s face (principal vector) with an angular position whose value is $\alpha_{p}(t)+\pi$ , while the angular position of cart $S$ has to move to point the patient’s body (secondary vector) with an angular position whose value is $\alpha_{e}(t)+\pi$ , where $\alpha_{e}(t)$ is computed as follows:

$\displaystyle\alpha_{e}(t)\!=\!\arctan\!{\!\left(\!\!\right)}$ (42)

Figure 11.

3D graphs of case study A: (a) starting configurations; (b) final configurations.

Figure 12.

Controlled evolution of cart $Q$ (left side) and cart $S$ (right side) in case study A: (a) angular position around the circle that the rail forms, $\beta_{Q}(t)$ (a1), $\beta_{S}(t)$ (a2); (b) the angular position of the pan of the RGB-D sensor mounted on cart, $\gamma_{Q}(t)$ (b1), $\gamma_{S}(t)$ (b2); (c) the angular position of the tilt of the RGB-D sensor mounted on cart, $\alpha_{Q}(t)$ (c1), $\alpha_{S}(t)$ (c2).

The data obtained for the study are included in Table 5. Figure 11 depicts the 3D initial and final configuration of the system showing that the system has correctly selected the route required to travel the shortest path for the angular positions of both carts’ tilt. In this case, the routes are clockwise and counter-clockwise for angular positions of the RGB-D sensors’ tilt of carts $Q$ and $S$ , respectively. As can be seen in Fig. 11b, the final angular position of cart $Q$ points directly towards the patient’s master vector $\mathbf{v_{p}}(t)$ , while the final angular position of cart $S$ is similarly pointing to observe the patient’s slave vector $\mathbf{v_{s}}(t)$ .

Table 5

Tilt according to path

Data (rad)	Cart $Q$	Cart $S$
Path clockwise	$-$ 0.9337	$-$ 6.2076
Path counter-clockwise	5.3495	$-$ 0.0755
Path selected	$-$ 0.9330	$-$ 0.0755

When grouping the aforementioned results, the reference trajectory vectors for carts $Q$ and $S$ are the following:

$\displaystyle\mathbf{x_{Q}^{*}}(t)=\mathbf{x_{MS}^{*}}(t)=\left[\begin{array}[% ]{c}1.1126\\ -1.1126\\ -0.9337\\ \end{array}\right]\ \ [\textit{rad}]$ (43) $\displaystyle\mathbf{x_{S}^{*}}(t)=\mathbf{x_{SL}^{*}}(t)=\left[\begin{array}[% ]{c}-1.0147\\ 0.2293\\ -0.0755\\ \end{array}\right]\ \ [\textit{rad}]$ (44)

Figure 13.

Input control evolution of carts $Q$ (left side) and cart $S$ (right side) in case study A: (a) torque produced by the motors of the cart, $T_{1Q}(t)$ (a1), $T_{1S}(t)$ (a2); (b) torque produced by the pan motor of the RGB-D sensor of the cart, $T_{2Q}(t)$ (b1), $T_{2S}(t)$ (b2); (c) torque produced by the tilt motor of the RGB-D sensor of the cart , $T_{3Q}(t)$ (c1), $T_{3S}(t)$ (c2).

Finally, Fig. 12 illustrates the controlled evolution of the angular positions of cart $Q$ , $\mathbf{x_{Q}}(t)$ , and cart $S$ , $\mathbf{x_{S}}(t)$ , using the standard PID control and the proposed GPI controller. In addition, Fig. 13 depicts the evolution of the torques produced by the motors and the RGB-D sensors of cart $Q$ , $\mathbf{T_{Q}}(t)$ and cart $S$ , $\mathbf{T_{S}}(t)$ . Both figures show that a good stabilisation and transient response and a correct positioning of the angular position of the carts are obtained. However, the GPI control has been found to have a better dynamic response than the PID control.

Figure 14.

Positioning graphs of the case study B: (a) starting configurations; (b) final configurations.

Figure 15.

3D graphs of the case B: (a) starting configurations; (b) final configurations.

Figure 16.

Controlled evolution of cart $Q$ (left side) and cart $S$ (right side) in case study B: (a) angular position around the circle that the rail forms, $\beta_{Q}(t)$ (a1), $\beta_{S}(t)$ (a2); (b) the angular position of the pan of the RGB-D sensor mounted on cart, $\gamma_{Q}(t)$ (b1), $\gamma_{S}(t)$ (b2); (c) the angular position of the tilt of the RGB-D sensor mounted on cart, $\alpha_{Q}(t)$ (c1), $\alpha_{S}(t)$ (c2).

Figure 17.

Input control evolution of cart $Q$ (left side) and cart $S$ (right side) in case study B: (a) torque produced by the motors of the cart, $T_{1Q}(t)$ (a1), $T_{1S}(t)$ (a2); (b) torque produced by the pan motor of the RGB-D sensor of the cart, $T_{2Q}(t)$ (b1), $T_{2S}(t)$ (b2); (c) torque produced by the tilt motor of the RGB-D sensor of the cart , $T_{3Q}(t)$ (c1), $T_{3S}(t)$ (c2).

Table 6

Performance of the control methods (case study A)

Control method	ISE	IAE	ITAE
GPI – Cart $Q$	0.4191	0.8782	0.5782
PID – Cart $Q$	0.5881	1.1199	0.8336
GPI – Cart $S$	0.3450	0.7237	0.4765
PID – Cart $S$	0.5129	0.9834	0.7471

Moreover, the performance of the control methods have been measured in terms of the integral squared tracking error, $\textit{ISE}=\int^{t_{B}}_{t_{A}}{e^{2}(t)\mathrm{d}t}$ , the integral absolute tracking error, $\textit{IAE}=\int^{t_{B}}_{t_{A}}{\left|e(t)\right|\mathrm{d}t}$ , and the integral time absolute tracking error, $\textit{ITAE}=\int^{t_{B}}_{t_{A}}{t\cdot\left|e(t)\right|\mathrm{d}t}$ , where $t_{A}=0[s]$ and $t_{B}=8[s]$ denote the initial and final time of the simulation. The ISE and the IAE criteria will handle all the tracking errors in a uniform ma-nner [51]. However, as time appears as a factor in the ITAE criterion, it will heavily penalise errors that occur late in time, but will ignore errors that occur early in time [52, 53]. The results achieved in terms of ISE, IAE and ITAE are depicted in Table 6, showing a better performance (smaller values) of the proposed GPI controller which is in accordance with the theoretical developments [54].

6.2 Case study B

This case study is the one previously represented in Fig. 7b, where the instantaneous position of the patient in the circular rail is $\mathbf{P}=$ [ $-$ 0.5, $-$ 0.5, 0.5] ${}^{T}$ , the instantaneous position of the part of the patient’s body to be monitored is $\mathbf{P_{b}}=$ [ $-$ 0.5, $-$ 0.5, 1.2] ${}^{T}$ , the direction in which the patient is facing is $\mathbf{v_{p}}=[-{\textstyle\frac{\sqrt{2}}{2}},{\textstyle\frac{\sqrt{2}}{2}}% ,0]^{T}$ , the direction in which the part of the patient’s body to be monitored is pointing, $\mathbf{v_{s}}=[-1,0,0]^{T}$ , and the initial angular positions of the two carts which are inside the circle traced by the rail are $\mathbf{x_{Q_{0}}}=[{\textstyle\frac{\pi}{2}},{\textstyle\frac{7\pi}{4}},0]^{T}$ [rad] and $\mathbf{x_{S_{0}}}=[{\textstyle\frac{\pi}{4}},{\textstyle\frac{5\pi}{4}},0]^{T}$ [rad], respectively.

The intelligent trajectory generator foresees a collision of both carts and selects the optimal movement of the carts (which is given by the minimum path travelled by the two carts) and immediately changes the roles of the carts, with cart $Q$ playing the role of slave and cart $S$ taking on the role of master, i.e. $\mathbf{x_{Q}^{*}}(t)=\mathbf{x_{SL}^{*}}(t)$ and $\mathbf{x_{S}^{*}}(t)=\mathbf{x_{MS}^{*}}(t)$ . The following reference trajectory vectors for carts $Q$ and $S$ were obtained as:

$\displaystyle\mathbf{x_{Q}^{*}}(t)=\mathbf{x_{SL}^{*}}(t)=\left[\begin{array}[% ]{c}1.8001\\ -1.0147\\ -0.5770\\ \end{array}\right]\ \ [\textit{rad}]$ (45) $\displaystyle\mathbf{x_{S}^{*}}(t)=\mathbf{x_{MS}^{*}}(t)=\left[\begin{array}[% ]{c}1.8980\\ -0.3272\\ -0.2356\\ \end{array}\right]\ \ [\textit{rad}]$ (46)

Figures 14 and 15 show the projection on the XY-plane and the 3D view of the initial and final configuration of the system, respectively. It is observed that the system has correctly selected the route required to travel the shortest path, demonstrating the effectiveness of the proposed intelligent trajectory planner and achieving that the final angular position of cart $Q$ points directly towards the patient’s slave vector $\mathbf{v_{s}}$ , while the final angular position of cart $S$ is similarly pointing to observe the patient’s master vector $\mathbf{v_{p}}$ .

On the other hand, Fig. 16 depicts the controlled evolution of the angular positions of cart $Q$ , $\mathbf{x_{Q}}(t)$ , and cart $S$ , respectively, using the standard PID control and the proposed GPI controller, while Fig. 17 illustrates the evolution of the torques produced by the motors and the RGB-D sensors of cart $Q$ , $\mathbf{T_{Q}}(t)$ and cart $S$ , $\mathbf{T_{S}}(t)$ .

Additionally, Table 7 shows the performance measures for the control methods. Again, the GPI control has been found to have a better dynamic response than the PID control and shows a better performance of the proposed control in terms of ISE, IAE and ITAE.

Table 7

Performance of the control methods (case study B)

Control method	ISE	IAE	ITAE
GPI – Cart $Q$	0.7320	1.0298	0.6780
PID – Cart $Q$	1.0855	1.3760	1.0400
GPI – Cart $S$	0.8029	1.0392	0.6842
PID – Cart $S$	1.1937	1.4122	1.0728

7. Conclusions

This paper has introduced an intelligent trajectory generator and a GPI control law for the motion of two carts, each equipped with an RGB-D sensor, on a circular motorised rail. The ultimate objective of the system described was a continuous monitoring of the facial emotion expressed by the patient and the physical rehabilitation exercises performed. This is thought for the evaluation of physical rehabilitation exercises by calculating possible deviations from the optimal gestures, while simultaneously assessing the patient’s degree of comfort during rehabilitation by monitoring his/her felt emotions.

The proposed feedback control scheme was based on an exact feed-forward action combined with a velocity vector structural estimation using a model-based integral state re-constructor that allowed us to achieve a trajectory tracking controller, guaranteeing to track the desired trajectories in an exponential asympto-tically stable nature. This was done without the use of state measurements, or without devising the dynamic systems normally employed to solve state estimation problems, as traditionally provided by asymptotic state observers.

Moreover, it was not necessary to use low-pass filters to improve the noise estimation of output time derivatives by means of sampling and discrete appro-ximations. Furthermore, the proposed intelligent trajectory generator comprised the references required to move the carts based on the instantaneous position of the patient inside the circular rail and the current configuration of the two carts. It also established a role (master or slave) for each cart, the direction that each cart had to follow, and generated smooth trajectories that avoided collisions between the carts.

The paper, therefore, has described the displacements of both carts and the positioning of the tilt and pan angles of each RGB-D sensor so as to track the patient’s face and the rehabilitated body part from the best viewpoints. In addition, the paper has also described a couple of case studies that demonstrate the effectiveness of the proposed approach, with a high performance of the feedback regulation scheme in the stabilisation and the trajectory tracking tasks of the carts, while security was maintained to ensure that both carts never would collide during their tracking of the person undergoing a rehabilitation program.

Given the results obtained in simulation, our future work will focus on the development of a real laboratory experimental platform. Tests will be performed to evaluate the performance of the trajectory planner and the GPI controllers of both carts in real physical rehabilitation scenarios. In addition, real images captured from RGB-D sensors will be processed to validate the current proposal.

Footnotes

Acknowledgments

This work was partially supported by Spanish Mini-sterio de Ciencia e Innovación, Agencia Estatal de Investigación (AEI) / European Regional Development Fund (FEDER, UE) under PID2019-106084RB-I00 and DPI2016-80894-R grants, by CIBERSAM of the Instituto de Salud Carlos III, and by VALU3S EU project (H2020-ECSEL grant agreement no 876852).

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Intelligent trajectory planner and generalised proportional integral control for two carts equipped with a red-green-blue depth sensor on a circular rail

Abstract

Keywords

1. Introduction

2. General description of the system

Table 1 List of symbols

5.1 Robust feed-forward controller for the motorised circular platform

Footnotes

Acknowledgments

References

Table 1
List of symbols