Modified Linear Technique for the Controllability and Observability of Robotic Arms

In this study, a modified linear technique is proposed for the controllability and observability of robotic arms, the modified linear technique consists of the following steps: a transformation is used to rewrite a nonlinear time-variant model as a linear time-variant model, this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, and the rank condition tests the controllability and observability of the linear time-invariant model. The modified linear technique is better than the linearization technique because the modified linear technique does not use the Jacobian approximation, while the linearization technique needs the Jacobian approximation. The modified linear technique is better than the linear technique because the modified linear technique can be applied to robotic arms with rotational and prismatic joints, while the linear technique can only be applied to robotic arms with rotational joints.


I. INTRODUCTION
T HE concept of controllability denotes the ability to move a robotic arm around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the kind of applied robotic arms [1], [2], [3]. Controllability and observability are dual aspects of the same problem.
Several authors have proposed interesting controllers or observers such as [4], [5], [6], [7]; however, in most of the studies, the robotic arms are assumed to be controllable or observable without any proof. A technique for the controllability of a robotic arm is important because a controllable robotic arm can guarantee the existence of a controller to reach one of the objectives such as the regulation, tracking, disturbance rejection, etc. [8], [9], [10], [11]. A technique for the observability of a robotic arm is important because an observable robotic arm can guarantee the existence of an observer to reach one of the objectives such as the states estimation, disturbance estimation, fault estimation, etc. Hence, it would be interesting to study when the controllability and observability of robotic arms are not assumed.
The rank condition for the controllability and observability of linear time-invariant models is the obvious and precise option, the starting point in the difficulty of this study is when nonlinear time-variant models such as the robotic arms are considered, in this case, the rank condition for the controllability and observability can not be directly applied to robotic arms. Thus, other alternative techniques to obtain this objective should be studied.
From the aforementioned studies, the linearization technique has been frequently applied for the controllability and observability of robotic arms [25], [26], [27], [28], [29], [30], [31], [32], [33], the linearization technique consists of the following three steps: 1) the Jacobian approximation is used to approximate a nonlinear time-variant model as a linear time-variant model, 2) this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, 3) the rank condition tests the controllability and observability of the linear time-invariant model. The linearization technique for the controllability and observability of robotic arms has the problem that it produces an undesired error caused by the Jacobian approximation. On the other hand, the linear technique also has been frequently applied for the controllability and observability of robotic arms [34], [35], [36], [37], [38], [39], [40], [41], the linear technique consists of the following three steps: 1) if all the joints in the robotic arm are rotational, then mathematical operations are used to rewrite the nonlinear time-variant model as a linear time-variant model, 2) this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, 3) the rank condition tests the controllability and observability of the linear timeinvariant model. The linear technique for the controllability and observability of robotic arms has the problem that it can not be applied to robotic arms with prismatic joints. It would be interesting to propose a technique for the controllability and observability of robotic arms which evade the problems of the linearization and linear techniques.
In this study, a modified linear technique is proposed for the controllability and observability of robotic arms, the modified linear technique consists of the following three steps: 1) a transformation is used to rewrite a nonlinear time-variant model as a linear time-variant model, 2) this linear timevariant model is evaluated at origin to obtain a linear timeinvariant model, 3) the rank condition tests the controllability and observability of the linear time-invariant model. Even the modified linear technique is similar to the linearization and linear techniques in the last step, it is very different in the first step, resulting on the following two advantages: 1) the modified linear technique is better than the linearization technique because the modified linear technique does not use the Jacobian approximation, while the linearization technique needs the Jacobian approximation, 2) the modified linear technique is better than the linear technique because the modified linear technique can be applied to robotic arms with rotational and prismatic joints, while the linear technique can only be applied to robotic arms with rotational joints.
Finally, the modified linear technique is compared with the linearization and linear techniques for the controllability and observability of the scara and two links robotic arms. The scara and two links robotic arms are selected because they could be applied in pick and place, screwed, printed circuits boards, packaging and labeling, etc. This paper is organized as follows. In section II, the linearization, the linear, and the modified linear techniques are detailed. In section III, the linearization, the linear, and the modified linear techniques are applied for the controllability and observability of the scara robotic arm. In section IV, the linearization, the linear, and the modified linear techniques are applied for the controllability and observability of the two links robotic arm. In section V, the conclusion and the forthcoming work are detailed.
In this section, the nomenclature of the robotic arms models is shown in Table I. VOLUME 4, 2016  joint angles or link displacements z 2 ∈ n×1 velocities W (z 1 ) ∈ n×n matrix with the inertia terms V (z 1 , z 2 ) ∈ n×n matrix with the Coriolis terms X(z 1 ) ∈ n×1 vector with the gravity terms 0 n ∈ n×1 vector of zeros I n×n ∈ n×n identity matrix 0 n×n ∈ n×n matrix of zeros z ∈ 2n×1 the states y ∈ n×1 the outputs F (z 1 , z 2 ) ∈ 2n×1 nonlinear function vector gravity acceleration m i ∈ mass of the link i l i ∈ length of the link i l ci ∈ half length of the link i J i ∈ moment of inertia of the link i

A. LINEARIZATION TECHNIQUE FOR THE CONTROLLABILITY AND OBSERVABILITY OF ROBOTIC ARMS
The linearization technique has been frequently applied for the controllability and observability of robotic arms [25], [26], [27], [28], [29], [30], [31], [32], [33], the linearization technique consists of the following three steps: 1) the Jacobian approximation is used to approximate a nonlinear time-variant model as a linear time-variant model, 2) this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, 3) the rank condition tests the controllability and observability of the linear time-invariant model. Consider the n-link planar robotic arm having n rotational and prismatic joints, the robotic arm model is: The fact that the derivative of the joint angles or link displacements z 1 ∈ n×1 is equal to the velocities z 2 ∈ n×1 is represented as follows: Using (2), the robotic arm model (1) is rewritten as follows: Considering (3), the nonlinear time-variant model (3) is: (4) is linearized at origin by the Jacobian approximation to obtain the following linear time-invariant model: Since the linear time-invariant model has been obtained in (5), the rank condition of linear systems [32], [33] is used for the controllability and observability of robotic arms.
Using the linear time-invariant model (5) of the linearization technique, the controllability matrix is: If the rank of the controllability matrix C 0 is equal to 2n, then the robotic arm of the linearization technique is controllable at origin. Using the linear time-invariant model (5) of the linearization technique, the observability matrix is: If the rank of the observability matrix O 0 is equal to 2n, then the robotic arm of the linearization technique is observable at origin.

Remark 1.
The linearization technique for the controllability and observability of robotic arms has the problem that it produces an undesired error caused by the Jacobian approximation.

B. LINEAR TECHNIQUE FOR THE CONTROLLABILITY AND OBSERVABILITY OF ROBOTIC ARMS
The linear technique also has been frequently applied for the controllability and observability of robotic arms [34], [35], [36], [37], [38], [39], [40], [41], the linear technique consists of the following three steps: 1) if all the joints in the robotic arm are rotational, then mathematical operations are used to rewrite the nonlinear time-variant model as a linear timevariant model, 2) this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, 3) the rank condition tests the controllability and observability of the linear time-invariant model.

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Since the linear time-invariant model has been obtained in (15), the rank condition of linear systems [32], [33] is used for the controllability and observability of robotic arms.
Using the linear time-invariant model (15) of the linear technique, the controllability matrix is: If the rank of the controllability matrix C 0 is equal to 2n, then the robotic arm of the linear technique is controllable at origin. Using the linear time-invariant model (15) of the linear technique, the observability matrix is: If the rank of the observability matrix O 0 is equal to 2n, then the robotic arm of the linear technique is observable at origin.
Remark 2. The linear technique for the controllability and observability of robotic arms has the problem that it can not be applied to robotic arms with prismatic joints.

C. MODIFIED LINEAR TECHNIQUE FOR THE CONTROLLABILITY AND OBSERVABILITY OF ROBOTIC ARMS
A modified linear technique is proposed for the controllability and observability of robotic arms, the modified linear technique consists of the following three steps: 1) a transformation is used to rewrite a nonlinear time-variant model as a linear time-variant model, 2) this linear time-variant model is evaluated at origin to obtain a linear time-invariant model, 3) the rank condition tests the controllability and observability of the linear time-invariant model. Consider the n-link planar robotic arm having n rotational and prismatic joints, the robotic arm model is: By using the fact that the inverse of the robotic arm inertia matrix W (z 1 ) ∈ n×n is well defined, the robotic arm model can be rewritten as: The fact that the derivative of the joint angles or link displacements z 1 ∈ n×1 is equal to the velocities z 2 ∈ n×1 is represented as follows: Using (19), (20), the nonlinear time-variant model is: Since the following transformation is true: We can apply the transformation (22) to the second equation of the nonlinear time-variant model (21) to obtain a term depending of z 1 ∈ n×1 as follows: Defining H(z 1 ) ∈ n×1 , the second equation of the nonlinear time-variant model (21) is rewritten as the following transformation: By using the transformation (24), the nonlinear timevariant model (21) is rewritten as the following linear timevariant model: The linear time-invariant model (25) is evaluated at origin to obtain the following linear time-invariant model: Since the linear time-invariant model has been obtained in (26), the rank condition of linear systems [32], [33] is used for the controllability and observability of robotic arms.
Using the linear time-invariant model (26) of the modified linear technique, the controllability matrix is: If the rank of the controllability matrix C 0 is equal to 2n, then the robotic arm of the modified linear technique is controllable at origin. Using the linear time-invariant model (26) of the modified linear technique, the observability matrix is: If the rank of the observability matrix O 0 is equal to 2n, then the robotic arm of the modified linear technique is observable at origin.

Remark 3. Even the modified linear technique is similar
to the linearization and linear techniques in the last step, it is very different in the first step, resulting on the following two advantages: 1) the modified linear technique is better than the linearization technique because the modified linear technique does not use the Jacobian approximation, while the linearization technique needs the Jacobian approximation, 2) the modified linear technique is better than the linear technique because the modified linear technique can be applied to robotic arms with rotational and prismatic joints, while the linear technique can only be applied to robotic arms with rotational joints.

III. SCARA ROBOTIC ARM
In this section, we compare the three techniques for the controllability and observability in the scara robotic arm.
The scara robotic arm has three degrees of freedom, it has two rotational joints and two links configured in horizontal position, it has one prismatic joint and one link configured in vertical position. We express the scara robotic arm of the Figure 2, where θ 1 , θ 2 , are the angles of the joints one, two in rad, l c3 is the length of the link three, in m.

FIGURE 2. Scara robotic arm
In this section, the nomenclature of the scara robotic arm is shown in Table II.
By using the linearization technique described in (5), the VOLUME 4, 2016 linear time-invariant model is: By using the linearization technique described in (6), the controllability matrix is: Since the rank of the controllability matrix is 6, the scara robotic arm is controllable. By using the linearization technique described in (7), the observability matrix is: Since the rank of the observability matrix is 6, the scara robotic arm is observable.

B. LINEAR TECHNIQUE
The linear technique can not be applied to the scara robotic arm because the linear technique can only be applied to robotic arms with rotational joints and the scara robotic arm has one prismatic joint.
By using the modified linear technique described in (25), the linear time-variant model is: By using the modified linear technique described in (26), the linear time-invariant model is: By using the modified linear technique described in (27), the controllability matrix is: Since the rank of the controllability matrix is 6, the scara robotic arm is controllable.
By using the modified linear technique described in (28), the observability matrix is: Since the rank of the observability matrix is 6, the scara robotic arm is observable.

D. COMPARISON OF RESULTS
The linear technique can not be applied to the scara robotic arm because the linear technique can only be applied to robotic arms with rotational joints and the scara robotic arm has one prismatic joint. The linearization and modified linear techniques can be applied to the scara robotic arm where we obtained in both that the scara robotic arm is controllable and observable. It proves that the linearization and modified linear techniques are satisfactory options for the controllability and observability of the scara robotic arm.

IV. TWO LINK ROBOTIC ARM
In this section, we compare the three techniques for the controllability and observability in the two link robotic arm. The two link robotic arm has two degrees of freedom, it has two rotational joints and two links configured in vertical position. We express the two link robotic arm of the Figure  3, where θ 1 , θ 2 are the angles of the joints one, two in rad.

FIGURE 3. Two link robotic arm
In this section, the nomenclature of the scara robotic arm is shown in Table III.
By using the linearization technique described in (4), the nonlinear time-variant model is: Where the robotic arm terms F 1 , F 2 , · · · , F 4 of F (z 1 , z 2 ) ∈ 4×1 are described as follows: By using the linearization technique described in (5), the linear time-invariant model is:

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By using the linearization technique described in (6), the controllability matrix is: Since the rank of the controllability matrix is 4, the two link robotic arm is controllable. By using the linearization technique described in (7), the observability matrix is: Since the rank of the observability matrix is 4, the two link robotic arm is observable.
By using the linear technique described in (8), the nonlinear time-variant model is: Where the inertia terms W 11 , W 12 , · · · , W 22 of W (z 1 ) ∈ 2×2 , the centripetal and Coriolis terms V 11 , V 12 , · · · , V 22 of V (z 1 , z 2 ) ∈ 2×2 , and the gravity terms X 1 , X 2 of X(z 1 ) ∈ 2×1 are described as follows: The other terms of W (z 1 ), V (z 1 , z 2 ), X(z 1 ) are zero, C 2 = cos(z 12 ), S 2 = sin(z 12 ). By using the linear technique described in (14), the linear time-variant model is: By using the linear technique described in (15), the linear time-invariant model is: By using the linear technique described in (16), the controllability matrix is: Since the rank of the controllability matrix is 4, the two link robotic arm is controllable. By using the linear technique described in (17), the observability matrix is: Since the rank of the observability matrix is 4, the two link robotic arm is observable.

C. MODIFIED LINEAR TECHNIQUE
Now, the modified linear technique of this study is applied to the two link robotic arm.
By using the modified linear technique described in (25), the linear time-variant model is: By using the modified linear technique described in (27), the controllability matrix is: Since the rank of the controllability matrix is 4, the two link robotic arm is controllable. By using the modified linear technique described in (28), the observability matrix is: Since the rank of the observability matrix is 4, the two link robotic arm is observable.

D. COMPARISON OF RESULTS
The linearization, linear, and modified linear techniques can be applied to the two link robotic arm where we obtained in all that the two link robotic arm is controllable and observable. It proves that the linearization, linear, and modified linear techniques are satisfactory options for the controllability and observability of the two link robotic arm.

V. CONCLUSION
In this study, the modified linear technique is proposed for the controllability and observability of robotic arms. Some authors proposed a linearization technique and other proposed a linear technique, while this study considered a technique which has similarities and differences with the other two techniques. The numerical results showed that the modified linear technique can be applied to the two robotic arms, while the linear technique can only be applied to one of the two robotic arms. The proposed technique could be applied to any of the conventional structures of robotic arms. Since this study is mainly focused in a technique for the controllability and observability of robotic arms; in the forthcoming work, the proposed technique will be used to obtain a controller and an observer.
MARCO ANTONIO ISLAS is a Ph.D. student in the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional in 2020. He