n-th Order Sensor Output to Control k-DoF Serial Robot Arms

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 n-th order sensor output can be used to control k-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 n 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.


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 n > 0. 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 k-DoF serial robot arm with dynamical inclusion of linear n-order sensors is investigated, showing that robot's properties with linear n-order sensors inclusion are invariant with respect to robot's theo-retical dynamics, provided that the solutions of the considered linear n-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 filterbased mechanisms have been implemented to obtain highquality 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 n -order differential equations, making them unreliable in the world of robotics.
Motivated by the fact that currently sensory systems of order n > 0 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?

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 n-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. Figure 1 shows a block control diagram with a negative feedback summing a reference signal (Ref) to get the tracking error (e) as the input for the control block which is the controller for the system Robot-Sensors. In Figure 1, τ is the control torque, q represents the actual robot's position identified by q * (i.e., q * → q), whileq represents the sensor's output. If sensors are of order zero, i.e., β q = q, where matrix β −1 includes the sensor's parameters of sensitiveness, the feedback of states can be taken directly from the sensor's output. Nevertheless, sensors of order n can be designed and/or included to control robotic systems. In such a case, the problem consists of recovering q from the informationq 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 n 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.

Robot's Model with Sensor System Inclusion
Let us consider the dynamic differential equation where β i ∈ ℝ k×k with i = 0, 1, ⋯, n represents the functional diagonal matrices, i.e., each element of β denoted as β line×column = 0 for all line ≠ column;q ∈ ℝ k is the sensor's response vector; q ∈ ℝ k is the vector for the actual positions of robot joints; Fðx, y, z, tÞ ∈ ℝ k represents a position and/or time dependent possible function; and k is the number of DoF of the robot.
For the case of linear sensors, (1) becomes Now, let us consider the following nonlinear robot dynamics where vector x represents the states of the electromechanical plant, i.e., x = ½χ 1 , χ 2 T ; vector f ðt, xÞ is a time and/or statespossible dependent function; and x 0 is the initial condition value at t = t 0 . 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 n-th order for the closed-loop feedback of the states if the mathematical structure of the solutionq of the differential equations that model sensor's behaviour (2) exists and is unique for all t ≥ t 0 , under the following assumptions: Assumption 1. Sensors provide the dynamic solution of (2) for all t ≥ t 0 , i.e., it is assumed that the value for the signalq is available to be used at time t ≥ t 0 by the closed-loop control system. Assumption 2. It is assumed that the diagonal matrices β i with i = 1, ⋯, n in (2) are known.
where L is called the Lipschitz constant. (See [6] among many others.) Theorem 1 (Picard-Lindeloff approach). For function f : D ⊂ ℝ r+1 ↦ ℝ r locally Lipschitz, and also for the set let H be a bound of kf ðt, xÞk in R, L the Lipschitz constant of f ðt, xÞ in R, and α = min fa, b/H, 1/Lg. Then, problem (6) with initial conditions has a solution, and it is unique in the interval ½t 0 , t 0 + α (in the Picard-Lindeloff approach). _ where f = ½ f 1 , f 2 T , x = ½χ 1 , χ 2 T , and ðχ 1 Þ 0 and ðχ 2 Þ 0 are the initial values for position and velocity, respectively, at t = t 0 .
Proof. It is assumed thatq exists, then, also the set exists, where Cð½t 0 , t 0 + α, ℝ k Þ is a Banach space with uniform normq and F 0 is closed in C.
According to (2) and control concept expressed in Figure 1, the identification process of q is where q * ∈ℝ k is the result of processingq and ε ∈ ℝ k is an identification error given by the Euclidian norm |q * − q | . Observe that q * ðt 0 Þ = β 0q ðt 0 Þ + ε, for simplicity β 0 = I and ε → 0, then, q * ðt 0 Þ → x 0 , and the problem is reduced to the existence of the set as the norm kq * ðtÞ − q * ðt 0 Þk is bounded by b. Let us define Γ : F → Cð½t 0 , t 0 + α, ℝ k Þ such that the image of the function q * ðtÞ is Γ q * ðtÞ defined by where we accomplished both Then, Γ is a contraction defined in a closed subset of a complete metric space, which implies that there exists a unique q * ∈ F such that Γq * ðtÞ = q * ðtÞ, according to the fixed-point theorem of Banach.
Thus, q * ðtÞ exists and is a unique solution of (6) for all t 0 ≤ t ≤ t 0 + α.

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: Figure 1: Block diagram of the closed-loop control with sensors inclusion.

Theorem 2. Dynamic properties of
which are the dynamic robot's model with actual output q, remain invariant with respect to the transformed form which is the robot with the n-th order sensor inclusion, where τ ∈ ℝ k is the torque vector, M ∈ ℝ k×k is the inertia matrix, matrix C ∈ ℝ k×k represents the Coriolis and centripetal enforces, G ∈ ℝ k is the gravitational vector, and F τ f 1 , ⋯F τ f k 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 by which are the sensor's gain matrices, the articular positions, and the articular velocities, respectively.
Proof. The proof starts with k = 3, but the intermediate results are given to make evident it is the extension to k = 2, 1, and by induction conclude generalization to (13)-(15), i.e., generalization for all k ≥ 1.
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 n.
As dynamics is a consequence of kinematics, let us consider the form of direct and inverse kinematics with n-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 O 3×3 is a 3 × 3 matrix for the orientation, X 3×1 is a 3 × 1 array for the position of the tool, and H 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 (j − 1)-th to the j-th link, through two translation and two rotation movements, as is presented in the following homogeneous transformation matrix, Journal of Sensors where θ, d, a, and α 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; x j , y j , and z j are the axes of the j-th coordinate frame; R z j−1 ,θ j represents the rotation matrix with respect to z j−1 ; T z j−1 ,d j is a translation matrix with respect to z j−1 ; T x j ,a j is a translation matrix with respect to x j ; and R x j ,α j is a rotation matrix with respect to x j (see [8][9][10][11][12][13] for a detailed procedure in obtaining (17)).
To feed equation (17) with the values of θ, d, a and α, 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 q 1 , q 2 , and q 3 , which represent angular displacements as is shown in Figure 2; l = ½l 1 , l 2 , l 3 T are the link lengths, which are illustrated in the right side of Figure 3.
A 3-DoF robot arm has j = 1, 2, 3, as such, substituting DH parameters in (17) with Thus, from the last column in (18) and including (2), direct kinematics is obtained for the robot with n-order sensor inclusion, as follows   Inverse kinematics. Solving equation (20) for q 1 , q 2 , and q 3 , inverse kinematics (see [14]) is obtained. This can be done involving the Moore-Penrose pseudoinverse Jacobian matrix J T ðJ J T Þ −1 and taking into account the chain rule _ X = J _ q, for the first time derivative of q, given by _ q = J T ðJ J T Þ −1 _ X, where J is the Jacobian matrix J = ð∂/∂qÞX (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 O 3×3 inside (16) are needed to obtain ϕ, γ, and ψ 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 ϕ, γ , and ψ, 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 sys-tem inclusion given by Table 2, concluding that kinematics is invariant to sensor system inclusion.

Robot's Dynamics.
Let us represent the total torque vector (τ) of the robot under consideration according to the Euler-Lagrange approach, as where M ∈ ℝ 3×3 is the inertia matrix, U ∈ ℝ 3 is the potential energy vector, and F τ f ∈ ℝ 3 is the friction force vector. Using the methodology detailed in [15][16][17] to obtain the torque for the j-th link, (24) is transformed to the following form where L ∈ ℝ 3 is the Lagrangian given by with K and U as the kinetic and potential energies, respectively, which according to the referenced methodology are calculated for this work as ½l c1 , l c2 , l c3 T are the inertia, mass, and length vectors, respectively, for the j-th centroid, and g = 9:81m/s 2 .

Journal of Sensors
Separating each element of vector (25) and calculating where d dt Defining the Coriolis forces and gravitational vectors, respectively, as and 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), (MðqÞ, Cðq, _ qÞ, and Gðq Þ), are obtained and given in the following form\scale95%{{ Mq ð Þ = where Cq where 7 Journal of Sensors Additionally, the friction vector symbolic form must 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 follows which proves dynamics invariance to sensor inclusion.

Extensive
Considerations. Even more, for a serial robot with 2-DoF, from Tables 1 to 2, line i = 3 must be removed and (16) becomes Gq , Also, when a 1-DoF is considered, from Tables 1 to 2, line i = 2, 3 must be removed and (16) becomes with 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 k-DoF for all k ≥ 1.
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 β i as a constant, the following identity should be used, in order to simplify the algebraic complications, And for the case to consider matrix β i = I, Remark 5. For initial conditions at the originqð0Þ = _ qð0Þ = _ qð0Þ = ⋯ = 0, the Laplace transform for sensors of the k-th order with β i = I, such that 2.3. Control Using n-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 n (the order of every sensor) and sensor's dynamics are known; this implies that ε → 0.
Claim 2. Supported by Theorems 2 and 3, and if q * → q, it can be claimed that equation (2) becomes the method to identify q fromq to control (14), which represents the robot with n -th order sensor inclusion. Thus, where e ∈ ℝ k is the tracking error, and X ref ∈ ℝ k represents the reference signal. The method is depicted in Figure 4, where the dashed squared block of the negative feedback represents (2). Note 1. Solutionq 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 secondorder 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.

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 n-th order sensors are included on it and when conditions (including assumptions) are accomplished.
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 where u pid ∈ ℝ 3 represents the value for the torque (control signal); K p ∈ ℝ 3 , K i ∈ ℝ 3 , and K d ∈ ℝ 3 are the proportional, integral, and derivative gains, respectively; and e = q − X ref ∈ ℝ 3 represents the tracking error of the control system, with X ref as the reference signal. For this example, the PID controller is tuned with K p = 1000, K i = 500, and K d = 400 for every one of the robot joints. The other considered technique is the following Sliding Mode Control (SMC) of order two, u smc = −r 1 sgn e ð Þ − r 2 sgn _ e ð Þ, where u smc ∈ ℝ 3 represents the control signal generated by the SMC; r 1 , r 2 ∈ ℝ 3 are the so-called twisting parameters; and sgn ð·Þ is the well-known sign function. For this example, the twisting controller is tuned with r 1 = 100 and r 2 = 50, for the first joint; r 1 = 150 and r 2 = 75, for the second joint; and, r 1 = 50 and r 2 = 25, 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.

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 n-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 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

12
Journal of Sensors 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.

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 n-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, ε → 0 (3) The dynamics of the electromechanical plant with n -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 k-DoF, this condition is always fulfilled Assumptions: (1) The value of the plant's outputq 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 β i with i = 1, ⋯, n 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 closedloop 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.