Abstract
Currently, zero-order sensors are commonly used as positioning feedback for the closed-loop control in robotics; thus, in order to expand robots’ control alternatives, other paths in sensing should be investigated more deeply. Conditions under which the -th order sensor output can be used to control -DoF serial robot arms are formally studied in this work. In obtaining the mentioned control conditions, the Pickard-Lindeloff theorem has been used to prove the existence and uniqueness of the robot’s mathematical model solution with order sensory systems included. To verify that the given conditions and claims guarantee controllability for both continuous-based and variable structure-based systems, two types of control strategies are used in obtaining simulation results: the conventional PID control and a second-order Sliding Mode control.
1. Introduction
Currently, sensors of order zero are the most extensively used positioning detectors in closed-loop control systems in industrial robotics. As a matter of fact, nowadays, robotic systems nearly exclusively use incremental and absolute encoders, resolvers, and high precision potentiometers. This is due to the fact that robotic controllers require delay-free information to achieve the desired control goals: accuracy, precision, repetitiveness, avoidance of tracking error, reduction of speed error, and so on.
An extended paradigm considers that positioning sensors of higher order than zero are not reliable at all. This consideration demotivates engineers for designing sensors of order . Nevertheless, the authors of this work consider that the theme should be sufficiently investigated to eliminate restrictive considerations which could lead in the future, to improve sensor systems in robotics.
In this work, the performance of a -DoF serial robot arm with dynamical inclusion of linear -order sensors is investigated, showing that robot’s properties with linear -order sensors inclusion are invariant with respect to robot’s theoretical dynamics, provided that the solutions of the considered linear -order sensors exist and are unique.
Interest in studying the use of higher-order sensor devices in the robotic industry has increased due to the need to implement more efficient, faster, and optimal control systems in both types of sensor systems concentrated in a robotic arm or systems with distributed sensors connected in a network and which may also be subject to adverse environmental conditions. However, the use of higher-order sensor systems inevitably introduces time delays in robotic control systems, including those delays from the controller to the actuator and the delays from the sensor to the controller.
On the other hand, some control methodologies have been proposed, ranging from sliding mode and adaptive techniques to manage uncertainties related to the knowledge of external dynamics and disturbances [1, 2]. However, these approaches always work well in lag-free environments.
To try to compensate for these delays, low pass filter-based mechanisms have been implemented to obtain high-quality signals and be able to build control loops for control mechanisms [3].
Mathematical scaffolding to ensure unique solutions in robotic device control algorithms including higher-order sensor systems is not widely studied.
Even more, alternative solutions to deal with nonlinear systems with unmeasurable states have been proposed, in [4] where an adaptive fuzzy output-feedback control scheme is designed and in [5] where a fuzzy logic system-based switched observer is constructed to approximate the unmeasurable states.
1.1. Motivation
Sensors can be designed using a very wide variety of physical principles (e.g., inductive, capacitive); however, many phenomena must be modelled using the -order differential equations, making them unreliable in the world of robotics.
Motivated by the fact that currently sensory systems of order are not included either in industrial or in other areas of robotics, the authors are interested in expanding the study of the control of serial mechanisms using sensors that may have the potential to improve in some way the current ones.
As such, the first question is: under which conditions a serial robot mechanism can be controlled by positioning sensors of order different from zero?
If this first question is satisfactorily solved, a second one comes to mind: how can we take advantage of the sensor output for control purposes?
1.2. Contribution of This Work
The main contribution of this work is to study conditions to make possible the control of serial robot arms, taking as the feedback for the closed-loop control the output of the -th order sensory systems. Also, a technique to obtain the mentioned feedback signal from the sensor’s output is proposed.
As formal proofs for claims presented in the text are provided, the methodology of this work is rigorous.
As a proof of concept, an application example is focused on the trajectory tracking control problem (for simplicity); however, using the first time derivative of the position, the robot’s joint speed control follows the same rules given herein.
1.3. Problem Statement
Figure 1 shows a block control diagram with a negative feedback summing a reference signal (Ref) to get the tracking error () as the input for the control block which is the controller for the system Robot-Sensors. In Figure 1, is the control torque, represents the actual robot’s position identified by (i.e., ), while represents the sensor’s output. If sensors are of order zero, i.e., , where matrix includes the sensor’s parameters of sensitiveness, the feedback of states can be taken directly from the sensor’s output. Nevertheless, sensors of order can be designed and/or included to control robotic systems. In such a case, the problem consists of recovering from the information provided by sensors, using a processing method for this task. In the following section, this problem is addressed formally.
The remainder of the document is organized as follows: in Section 2, conditions and assumptions to control serial robots using the output of order sensory systems are given, and formal proofs for the given claims are presented; in Section 3 an implementation example for a 3-DoF serial robot arm is performed, along with simulation results; and finally, concluding remarks are given in Section 4.
2. Robot’s Model with Sensor System Inclusion
Let us consider the dynamic differential equationwhere with represents the functional diagonal matrices, i.e., each element of denoted as for all ; is the sensor’s response vector; is the vector for the actual positions of robot joints; represents a position and/or time dependent possible function; and is the number of DoF of the robot.
For the case of linear sensors, (1) becomes
Now, let us consider the following nonlinear robot dynamicswhere vector represents the states of the electromechanical plant, i.e., ; vector is a time and/or states-possible dependent function; and is the initial condition value at .
The inclusion of (2) in (3) is studied in this work, in order to obtain control conditions, according to the following claim:
Claim 1. It is possible the control of serial robots using positioning sensors of the -th order for the closed-loop feedback of the states if the mathematical structure of the solution of the differential equations that model sensor’s behaviour (2) exists and is unique for all , under the following assumptions:
Assumption 1. Sensors provide the dynamic solution of (2) for all , i.e., it is assumed that the value for the signal is available to be used at time by the closed-loop control system.
Assumption 2. It is assumed that the diagonal matrices with in (2) are known.
2.1. Existence and Uniqueness
Herein, the existence and uniqueness of the solution for the robot’s dynamics are verified for the case in which the solution of the sensor’s dynamics exists and is unique.
Definition 1 (Lipschitz). The function is locally Lipschitz if it is continuous, and for all subset compact exists such that if and , then,where is called the Lipschitz constant. (See [6] among many others.)
Theorem 1 (Picard-Lindeloff approach). For function locally Lipschitz, and also for the setlet be a bound of in , the Lipschitz constant of in , and . Then, problem (6) with initial conditions has a solution, and it is unique in the interval (in the Picard-Lindeloff approach).where , , and and are the initial values for position and velocity, respectively, at .
Proof. It is assumed that exists, then, also the setexists, where is a Banach space with uniform normand is closed in .
According to (2) and control concept expressed in Figure 1, the identification process of iswhere is the result of processing and is an identification error given by the Euclidian norm . Observe that , for simplicity and , then, , and the problem is reduced to the existence of the setas the norm is bounded by .
Let us define such that the image of the function is defined bywhere we accomplished both
Then, is a contraction defined in a closed subset of a complete metric space, which implies that there exists a unique such that , according to the fixed-point theorem of Banach.
Thus, exists and is a unique solution of (6) for all .
2.2. Dynamics Invariance
The validity of Claim 1 is conditioned to result in the transformation of the robot’s dynamics when the sensor’s dynamics are included. To solve this problem, the following theorem is presented:
Theorem 2. Dynamic properties ofwhich are the dynamic robot’s model with actual output , remain invariant with respect to the transformed formwhich is the robot with the -th order sensor inclusion, where is the torque vector, is the inertia matrix, matrix represents the Coriolis and centripetal enforces, is the gravitational vector, and , can be obtained by any parametric identification method, and each of them is the sum of viscous, Coulomb, and static friction forces.
The transformation method is given by the sensor inclusion in the robot’s joints, represented bywhich are the sensor’s gain matrices, the articular positions, and the articular velocities, respectively.
Proof. The proof starts with , but the intermediate results are given to make evident it is the extension to , and by induction conclude generalization to (13)–(15), i.e., generalization for all .
Let us consider a serial 3-DoF anthropomorphic robot arm as defined in [7] and as illustrated by Figure 2 with positioning sensors of order .
As dynamics is a consequence of kinematics, let us consider the form of direct and inverse kinematics with -th order sensor inclusion.
Direct kinematics. The Denavit-Hartenberg (DH) convention (see [8, 9]) is a commonly used method for selecting frames of reference and the relationship between them to obtain the kinematics mathematical model for robotic plants. This method uses arrays as those given by (16) to represent the orientation and the positioning of the end effector (i.e., the tool of the robot) [10, 11].where is a matrix for the orientation, is a array for the position of the tool, and is the so-called homogeneous transformation matrix (see [12]).
The DH convention claims that it is possible to obtain both position and orientation, from the ()-th to the -th link, through two translation and two rotation movements, as is presented in the following homogeneous transformation matrix,
where , , , and are the DH parameters, and their values come from specific aspects of the geometric relationship between the coordinate frames for the mechanical configuration of the robot under study and the way to select each of the coordinate frames; , , and are the axes of the -th coordinate frame; represents the rotation matrix with respect to ; is a translation matrix with respect to ; is a translation matrix with respect to ; and is a rotation matrix with respect to (see [8–13] for a detailed procedure in obtaining (17)).
To feed equation (17) with the values of , , and , Table 1 was constructed with the DH parameters of the robot under consideration, taking into account Figure 3 which shows the way in which the author has selected each of the coordinate frames (see [13] for more details).
Table 1 shows the articular plant positions , , and , which represent angular displacements as is shown in Figure 2; are the link lengths, which are illustrated in the right side of Figure 3.
A 3-DoF robot arm has , as such, substituting DH parameters in (17), three homogeneous transformation matrices are obtained: , , and . To go from system 0 to system 3, we compute , obtaining the final position and orientation of the tool
with
Thus, from the last column in (18) and including (2), direct kinematics is obtained for the robot with -order sensor inclusion, as followswhere , , and are the spatial Cartesian coordinates of the end effector.
Inverse kinematics. Solving equation (20) for , , and , inverse kinematics (see [14]) is obtained. This can be done involving the Moore-Penrose pseudoinverse Jacobian matrix and taking into account the chain rule , for the first time derivative of , given by , where is the Jacobian matrix (see [12]). Carrying out this procedure and involving (2), the inverse kinematics for the robot is given by
Attitude in the orientation of the end effector. Three elements of the submatrix inside (16) are needed to obtain , , and which represent the roll, pitch, and yaw movements of the end effector, respectively (see [13]):
Now, taking (18) as the master matrix and solving for , , and , the orientation of the end effector of 3-DoF anthropomorphic robot arms with any order linear positioning sensors is obtained as
The same results presented in (20)–(23) are obtained including directly in (18) the DH parameters with sensor system inclusion given by Table 2, concluding that kinematics is invariant to sensor system inclusion.
2.2.1. Robot’s Dynamics
Let us represent the total torque vector () of the robot under consideration according to the Euler-Lagrange approach, aswhere is the inertia matrix, is the potential energy vector, and is the friction force vector.
Using the methodology detailed in [15–17] to obtain the torque for the -th link, (24) is transformed to the following formwhere is the Lagrangian given bywith and as the kinetic and potential energies, respectively, which according to the referenced methodology are calculated for this work as where , , and are the inertia, mass, and length vectors, respectively, for the -th centroid, and .
Separating each element of vector (25) and calculatingwhere
Defining the Coriolis forces and gravitational vectors, respectively, asand including (2) in (24), the torque vector takes the form
Rearranging results presented in (29) and (30) and also including (2), the elements of (32), (, , and ), are obtained and given in the following formwherewhere
Additionally, the friction vector symbolic formmust be included in the dynamic model.
Even more, the same results presented in (33)–(39) are obtained, when (2) is directly included in the Euler-Lagrange (24) and in (31), as followswhere
which proves dynamics invariance to sensor inclusion.
2.2.2. Extensive Considerations
Even more, for a serial robot with 2-DoF, from Tables 1 to 2, line must be removed and (16) becomeswith
Also, when a 1-DoF is considered, from Tables 1 to 2, line must be removed and (16) becomeswith
Since procedures (16)–(40) have intermediated outcomes, it is easy to compute results for a 1-DoF and for a 2-DoF robot, to obtain the same conclusions.
Further, invoking the induction method, these results can be extended to serial robots with -DoF for all .
In order to deal with sensor inclusion mathematical complexity, the following remarks are useful for practical implementations:
Remark 4. For the case to consider matrix as a constant, the following identity should be used, in order to simplify the algebraic complications,
And for the case to consider matrix ,
Remark 5. For initial conditions at the origin , the Laplace transform for sensors of the -th order with ,such that
2.3. Control Using -th Order Sensors’ Output
Now, in order to take advantage of Claim 1, the following control conclusions are given:
Assumption 3. It is assumed that sensors are previously well characterized, i.e., both (the order of every sensor) and sensor’s dynamics are known; this implies that .
Claim 2. Supported by Theorems 2 and 3, and if , it can be claimed that equation (2) becomes the method to identify from to control (14), which represents the robot with -th order sensor inclusion. Thus,where is the tracking error, and represents the reference signal.
The method is depicted in Figure 4, where the dashed squared block of the negative feedback represents (2).
Note 1. Solution of (2) cannot be directly used to control (6).
Extension of Claim 2. In [18], the Lasalle-invariance principle (generally used to show asymptotic stability in the Lyapunov approach) is extended to nonautonomous switching systems. This implies that Claim 2 is valid using both continuous-based and discontinuous-based controllers, e.g., Proportional Integral Derivative (PID) control and Sliding Mode (SM) controllers.
Even more, in [19], an extensive study to show the control of MIMO systems by the twisting-algorithm (a second-order SM control) is presented, showing asymptotic convergence of the system trajectories to the selected sliding manifold.
Further, in order to review the stability for biorder SM (twisting-algorithm included) control with variable PID gains, [20] can be consulted.
3. Implementation Example
To verify the validity of Claims 1 and 2, this example is designed to solve the trajectory tracking control of the anthropomorphic robot with 3-DoF (see Figures 2, 3, and 5) when -th order sensors are included on it and when conditions (including assumptions) are accomplished.
Selecting , and linear sensors, with constant coefficients of order two, zero, and one, are included in joints one, two, and three, respectively, as
Reference signal to track for the robot is selected as
Thus, the ideal form of sensor systems is
Solving for , for the open-loop system, the solution has the structure
As the second and third terms in are negative for and they are positive for , then, the form of solution exists and is unique.
Note 2. Equations (55)–(56) are not included in the control method; they just show the ideal solution structure. For closed-loop control purposes, are obtained dynamically from sensor’s output.
Even more, for selecting parameters of Table 3 and friction models, the actual robot shown in Figure 5 was used, but no other application was performed using the real robot.
The following friction values are used:where it is possible to see that if any articular velocity of the robot becomes zero, then the corresponding friction equation is reduced to zero; the sign function (sgn) appears in the second terms. This vector includes friction forces produced by ball bearings, gear boxes, and drive belts. Thus, the approximated values can be obtained directly from the manufacturer’s mechanical components.
Substituting the parameters of Table 3 in (33)–(39) considering the friction model, and including sensors (52), numerical values for , , , and are given bywherewith the following numerical values
3.1. Controllers Used
Two techniques with different control properties are considered in this example to solve the trajectory tracking problem of the considered robot: the classical Proportional Integral Derivative (PID), and a Variable Structure System-based method.
The PID technique produces a smooth control signal, given by,where represents the value for the torque (control signal); , , and are the proportional, integral, and derivative gains, respectively; and represents the tracking error of the control system, with as the reference signal. For this example, the PID controller is tuned with , , and for every one of the robot joints.
The other considered technique is the following Sliding Mode Control (SMC) of order two,where represents the control signal generated by the SMC; are the so-called twisting parameters; and is the well-known sign function. For this example, the twisting controller is tuned with and , for the first joint; and , for the second joint; and, and , for the third joint.
Thus, the whole control system for this implementation example is depicted in Figure 6, where the reference signal (54), the control either (65) or (66), the robot (58)–(64), and the sensors (52)–(53), along with the closed-loop feedback given by Claim 2, are included.
3.2. Simulation Results
It is worth to mention that this section is not intended to show the benefits of the two control techniques presented, but to show that the robot with -th order sensor inclusion has controllability properties.
Figures 7, 8, 9 show the performance of the first, second, and third joints of the robot in tracking a square wave form, a sine wave, and an upward sloping function, respectively (in dashed red line), under the control of the PID (continuous black line) and under the SMC (continuous blue line).
In these simulations, the characteristic behaviour of both controllers can be appreciated, which supports that Claim 2 is correct. The PID is tuned with high gain values to obtain fast convergence and also to avoid large overshoots. Nevertheless, the PID comes out from convergence when an adjacent joint suddenly changes, i.e., the sudden change of a joint impacts (or even shocks) the convergence of the adjacent ones. Also, it can be appreciated that the SMC is a robust control technique, because it never comes out from convergence once it is reached (after the transient state).
Even more, Figure 10 shows an enlarged view of Figure 7 from 6 to 6.5 [s], where the so-called chattering effect produced by the SMC appears.
4. Conclusions
As a result of this work, it is possible to affirm that serial robots can be controlled by taking advantage of the output of -th order sensory systems, provided the following conditions and assumptions are fulfilled.
Conditions:(1)The solution of the differential equations that describe the dynamics of individual sensors must exist and be unique(2)The identification value of the actual robot’s position should be negligible, (3)The dynamics of the electromechanical plant with -th order sensor inclusion should be invariant with respect to the dynamics of the plant’s output. In this work, it has been demonstrated that for the case of serial robot manipulators with -DoF, this condition is always fulfilled
Assumptions:(1)The value of the plant’s output is always available to be used by the closed-loop control system(2)Coefficients of terms that describe the sensor dynamics are known. In this work, these terms are represented by diagonal matrices with in (2).(3)Plant’s dynamics is known, i.e., the differential equation that model plant’s dynamics is characterized
Then, serial robot manipulators are controllable using either continuous or variable structure-based controllers, and the differential equation that models sensor dynamics becomes the method to obtain the feedback for the closed-loop control.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Special thanks are due to Instituto Tecnológico José Mario Molina Pasquel y Henriquez, Lagos de Moreno, and Universidad de Guadalajara, CULagos, for their support to this work.