On the coupler point velocity control of variable input speed servo-controlled four-bar mechanism

Abstract

In this contribution, a four-bar linkage having a variable input velocity is studied, traditionally it is assumed to be constant. The advantages of a variable input velocity mechanism, in contrast to a mechanism driven by constant velocity, are the flexibility of the output motion (and/or improved kinematic and dynamic characteristics). The velocity of the crank is controlled in order to obtain the desired output motion at the coupler point. The input velocity trajectory and the controller parameters are considered as design variables, such that the kinematic and dynamic requirements are fulfilled. Two numerical examples are provided to corroborate the result.

Keywords

Variable input velocity four-bar linkages mechanism control velocity control crankshaft

Introduction

Traditionally, the dimensional synthesis of first-order mechanisms comprises the problems of path, function, and motion generation which is known as finite synthesis. However, under certain conditions, it is required to consider in the design of mechanisms the velocity and/or acceleration as requirements to be met.¹ When velocities and/or accelerations are considered in the design problem, the synthesis is known as higher-order synthesis.² There are many situations in the design of mechanical devices in which it is necessary to deal with constraints of velocity. For example, there is a real need for straight-line motions in machinery of all kinds, especially in automated production machinery. Economic considerations continually demand higher production rates, requiring higher speeds or additional expensive machines. When the product is in continuous motion in a straight line and at constant velocity, every workhead that operates on the product must be articulated to chase the product and match both its straight-line path and its constant velocity while performing the task.³ Several works have been published dealing with the mechanism design with velocity targets.

In particular, the higher-order synthesis for function generation has been studied in Ángeles Álvarez,¹ Erdman and Sandor,² and Sandor et al.⁴ The optimal design of a four-bar linkage for path generation task where a coupler point must satisfy certain conditions for position and velocity was performed in Norton,³ Schaefer and Kramer,⁵ Holte et al.,⁶ Robson and McCarthy,⁷ Ureña et al.,⁸ De-Juan et al.,⁹ and Russell and Shen.¹⁰ One of the basic assumptions in both the synthesis and analysis of the four-bar mechanism in many design cases is that the angular velocity of the input link is constant. This assumption is essential for the design of a mechanism in which timing requirements are involved. However, this assumption may not be valid if an electric motor is connected to drive the mechanism.¹¹ Moreover, in all above works, the problem of fulfilling the desired velocity profile was addressed from a dimensional synthesis point of view; this solution fulfills the velocity requirements only at the designs points and not in all the trajectory. If the desired velocity profile is changed and we are in an early stage of design mechanisms, like kinematic synthesis, it is relatively easy to calculate the new parameters of the mechanism that meet the requirements of the desired task. However, if the mechanism is already operational and it is imperative to work with a new operating speed, for example, the required parameters of production speed vary, a complete redesign of the mechanism would be necessary.

Recently, an alternative approach for achieving this purpose without modifying the geometric dimensions of the existing mechanism was presented in Yang et al.,¹² where was developed and designed a model of variable input speed of a compound mechanism with a slider-crank and a screw mechanism for improving the kinematic and dynamic characteristics. Here, Bezier functions are used as the desired velocity profile. Also, a novel concept of a four-bar mechanism equipped with a continuously variable transmission (CVT) has been proposed to obtain variable input speeds is presented in Yildiz et al.¹³ Thus, different manufacturing operations and transportations can be performed with the same mechanism without using an electrical inverter or changing the electrical motor. But in these approaches above was employed another mechanism to drive an existing mechanism to improve it.

In the present work, we are interested in moving the coupler point with a constant velocity, avoiding the mechanism redesign or using another mechanism to drive the existing mechanism, by controlling the angular velocity of the input link. This input desired velocity profile is achieved using a proportional–integral–derivative (PID) controller.

The equation of motion and mathematical model for the motor and PID controller are established to analyze the dynamic response of a servo-motor-driven four-bar linkage for a variable input speed. In this way, a constant speed in the coupler point over the complete cycle or range of motion can be reached.

This article is organized as follows: in section “Input speed profile,” the input profile angular velocity function is presented in order to fulfill the velocity target. The mathematical model of the motor-driven mechanism system is presented in section “Mathematical model of the system.” In section “Velocity controller,” we analyze the stability of the closed-loop system consisting of the PID controller. Illustrative examples for the problem of velocity tracking, to demonstrate the effectiveness of the control scheme, are shown in section “Simulation results.” Finally, some conclusions and future developments are exposed in section “Conclusion.”

Input speed profile

The advantages of a variable input speed mechanism, in contrast to a mechanism that is driven with constant speed, are the flexibility of the output motion and improved kinematic and dynamic characteristics.¹⁴ In what follows, the representation of the input speed trajectory is derived.

Coupler point velocity

The position of point P (see Figure 1) viewed from the inertial frame is given in equation (1)

\vec{P} = {\vec{r}}_{0} + R (\hat{z}, α)^{r} \vec{P}

(1)

where $R (\hat{z}, α)$ is the matrix that describes the rotation of an angle $α$ about axis $\hat{z}$

R (\hat{z}, α) = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]

(2)

Figure 1.

General four-bar linkage.

Now, replacing equation (2) in equation (1) results in

\begin{matrix} \vec{P} = \begin{matrix} [\begin{matrix} x_{0} \\ y_{0} \end{matrix}] + [\begin{matrix} L_{2} \cos (ϕ_{2} + α) + r_{cx} \cos (ϕ_{3} + α) \\ L_{2} \sin (ϕ_{2} + α) + r_{cx} \sin (ϕ_{3} + α) \end{matrix} \\ \begin{matrix} - r_{cy} \sin (ϕ_{3} + α) \\ + r_{cy} \cos (ϕ_{3} + α) \end{matrix}] \end{matrix} \end{matrix}

(3)

The linear velocity for the point P can be derived by continuously differentiating equation (3) with respect to time as follows

\begin{matrix} {\vec{V}}_{p} = \begin{matrix} [\begin{matrix} - L_{2} \sin (ϕ_{2} + α) {\overset{\cdot}{ϕ}}_{2} - r_{cx} \sin (ϕ_{3} + α) {\overset{\cdot}{ϕ}}_{3} \\ L_{2} \cos (ϕ_{2} + α) {\overset{\cdot}{ϕ}}_{2} + r_{cx} \cos (ϕ_{3} + α) {\overset{\cdot}{ϕ}}_{3} \end{matrix} \\ \begin{matrix} - r_{cy} \cos (ϕ_{3} + α) {\overset{\cdot}{ϕ}}_{3} \\ - r_{cy} \sin (ϕ_{3} + α) {\overset{\cdot}{ϕ}}_{3} \end{matrix}] \end{matrix} \end{matrix}

(4)

where ${\overset{\cdot}{ϕ}}_{2}$ is the input speed of the crank link and ${\overset{\cdot}{ϕ}}_{3}$ is the angular speed of coupler link given by

{\overset{\cdot}{ϕ}}_{3} = γ_{3} {\overset{\cdot}{ϕ}}_{2}

and $γ_{3}$ is defined in what follows

γ_{3} = \frac{L_{2} \sin (ϕ_{4} - ϕ_{2})}{L_{3} \sin (ϕ_{3} - ϕ_{4})}

Desired angular velocity profile function

From equation (4), we can obtain the linear velocity magnitude for the point P as follows

∥ {\vec{V}}_{p} ∥ = \sqrt{V_{px}^{2} + V_{py}^{2}}

(5)

As can be noted, if $∥ {\vec{V}}_{p} ∥$ is a function of ${\overset{\cdot}{ϕ}}_{2}$ , then it is possible to solve equation (5) in order to derive an input speed profile that maintains the coupler point velocity at a desired magnitude (6)

ω_{2} = \sqrt{\frac{V_{p x}^{2} + V_{p y}^{2}}{L_{2}^{2} + r_{c x}^{2} γ_{3}^{2} + r_{c y}^{2} γ_{3}^{2} + 2 L_{2}^{2} γ_{3} (r_{c x} \cos (ϕ_{2} - ϕ_{3}) + r_{c y} \sin (ϕ_{2} - ϕ_{3}))}}

(6)

Mathematical model of the system

Dynamics of the four-bar linkages

Figure 1 represents a general four-bar linkage. Link 2 (crank) is driven by an electrical motor, and it is able to perform a complete rotation. Link 3 (coupler) performs a general motion in the plane, and it transmits the movement to link 4 (rocker), which executes an oscillatory motion. In Figure 2, the center of mass for each body is denoted by a dark circle, and their locations are described by $r_{i}$ and $θ_{i}$ . For link i, $m_{i}$ represents the mass of each link, $J_{i}$ is the mass moment of inertia with respect to the centroid, and $L_{i}$ is the length for each link. The angular positions for the links with respect to the horizontal axis are denoted by $ϕ_{2}$ , $ϕ_{3}$ , and $ϕ_{4}$ .

Figure 2.

Four-bar mechanism.

The Lagrange’s equation can be written to obtain the equations of motion for the mechanism as

T = \frac{d}{dt} (\frac{\partial K}{\partial {\overset{\cdot}{ϕ}}_{2}}) - \frac{\partial K}{\partial ϕ_{2}} + \frac{\partial P}{\partial ϕ_{2}} + \frac{\partial D}{\partial {\overset{\cdot}{ϕ}}_{2}}

(7)

where $K, P, and D$ denote, respectively, the kinetic, potential, and dissipated energy, and T represents the externally applied torque. In equation (7), the crank angle $ϕ_{2}$ is used as the generalized coordinate describing the motion of the linkage. The kinetic energy is given by

K = \sum_{i = 2}^{4} [\frac{1}{2} m_{i} (V_{ix}^{2} + V_{iy}^{2}) + \frac{1}{2} J_{i} {\overset{\cdot}{ϕ}}_{i}^{2}]

(8)

where $V_{ix}$ and $V_{iy}$ represent the velocity components of the center of mass for link i, and ${\overset{\cdot}{ϕ}}_{4}$ is given by

{\overset{\cdot}{ϕ}}_{4} = γ_{4} {\overset{\cdot}{ϕ}}_{2}

with $γ_{4}$ defined as

γ_{4} = \frac{L_{2} \sin (ϕ_{3} - ϕ_{2})}{L_{4} \sin (ϕ_{3} - ϕ_{4})}

After an algebraic manipulation, equation (8) can be written as

K = \frac{1}{2} A (ϕ_{2}) {\overset{\cdot}{ϕ}}_{2}^{2}

(9)

where

A (ϕ_{2}) = c_{0} + c_{1} γ_{3}^{2} + c_{2} γ_{4}^{2} + c_{3} γ_{3} \cos (ϕ_{2} - Θ_{3})

(10)

with

\begin{matrix} c_{0} = J_{2} + m_{2} r_{2}^{2} + m_{3} L_{2}^{2} \\ c_{1} = J_{3} + m_{3} r_{3}^{2} \\ c_{2} = J_{4} + m_{4} r_{4}^{2} \\ c_{3} = 2 L_{2} r_{3} m_{3} \\ Θ_{3} = θ_{3} + ϕ_{3} \end{matrix}

The potential energy of the four-bar mechanism is due to the gravity, and it can be expressed as

\begin{array}{l} P = [m_{2} r_{2} \sin (θ_{2} + ϕ_{2}) + m_{3} (L_{2} \sin ϕ_{2} + r_{3} \sin (θ_{3} + ϕ_{3})) \\ + m_{4} (L_{1} \sin θ_{1} + r_{4} \sin (ϕ_{4} + θ_{4}))] g \\ \frac{\partial P}{\partial ϕ_{2}} = [m_{2} r_{2} \cos (θ_{2} + ϕ_{2}) + m_{4} (r_{4} γ_{4} \cos (θ_{4} + ϕ_{4})) \\ + m_{3} (L_{2} c o s ϕ_{2} + r_{3} γ_{3} \cos (θ_{3} + ϕ_{3}))] g \end{array}

(11)

The dissipative energy for the four-bar mechanism can be neglected, because there is no viscous torque present.

Replacing energy equations (8) and (11) in equation (7), the mechanical torque of the system is obtained as

\begin{matrix} T = A (ϕ_{2}) {\overset{\cdot\cdot}{ϕ}}_{2} + \frac{1}{2} \frac{\partial (A (ϕ_{2}))}{\partial ϕ_{2}} {\overset{\cdot}{ϕ}}_{2}^{2} + [m_{2} r_{2} \cos (θ_{2} + ϕ_{2}) \\ + m_{3} (L_{2} \cos ϕ_{2} + r_{3} γ_{3} \cos (θ_{3} + ϕ_{3})) + m_{4} (r_{4} γ_{4} \cos (θ_{4} + ϕ_{4}))] g \end{matrix}

(12)

Mathematical model for the mechatronic system

A schematic diagram of the electric motor is demonstrated in Figure 3. The gear ratio is given as

n = \frac{T_{b}}{T_{a}} = \frac{ω_{a}}{ω_{b}}

(13)

where $ω_{a}$ and $T_{a}$ are the angular velocity and the torque of the shaft a, respectively; $ω_{b}$ is the angular velocity of the shaft b; and $T_{b}$ is the output system and is equal to the mechanical torque of the system T defined in equation (12).

Figure 3.

Schematic diagram of a motor and a gearbox.

Using Kirchoff’s voltage law and Newtonian’s equation for the mechanism load, we have

V_{a} = Ri (t) + L \overset{\cdot}{i} (t) + e

(14)

T = n (T_{m} - T_{L} - B ω_{a} - J {\overset{\cdot}{ω}}_{a})

(15)

where $T_{L}$ is the load torque, the generated electromotive force given by $e = K_{g} ω_{a}$ and the magnetic torque $T_{m} = K_{m} i (t)$ , where $K_{m}$ and $K_{g}$ represent the constants for motor torque and motor voltage, respectively.

Equation (13) and $T_{m}$ can be substituted into equations (14) and (15); therefore, the mathematical model for the motor is as follows

\overset{\cdot}{i} (t) = \frac{1}{L} (V_{a} - Ri (t) - n K_{g} {\overset{\cdot}{ϕ}}_{2})

(16)

T = n K_{m} i (t) - n T_{L} - n^{2} B {\overset{\cdot}{ϕ}}_{2} - n^{2} J {\overset{\cdot\cdot}{ϕ}}_{2}

(17)

Now, the torque produced by the motor should be equal to the torque needed for the mechanical system, yielding this to the nonlinear equation of motion. In order to determine this equation, one can match equation (12) with equation (17), neglecting the terms related with the potential energy in equation (12) due to the orientation of operation of the mechanism, the resulting expression is

A (ϕ_{2}) {\overset{\cdot\cdot}{ϕ}}_{2} + \frac{1}{2} \frac{\partial (A (ϕ_{2}))}{\partial ϕ_{2}} {\dot{ϕ}}_{2}^{2} = n K_{m} i (t) - n T_{L} - n^{2} B {\dot{ϕ}}_{2} - n^{2} J {\overset{\cdot\cdot}{ϕ}}_{2}

(18)

In order to obtain a dynamical system including the mechanical and electrical dynamic, let us define the change of variables $x_{1} = ϕ_{2}$ , $x_{2} = {\overset{\cdot}{x}}_{1} = {\overset{\cdot}{ϕ}}_{2}$ , and $x_{3} = i (t)$ , and then from equations (16) and (18), the state-space representation of the whole system is as follows

{\overset{\cdot}{x}}_{1} = x_{2}

(19)

{\overset{\cdot}{x}}_{2} = A_{0} (A_{1} x_{2}^{2} + A_{2} x_{2} + n K_{m} x_{3} + A_{3})

(20)

{\overset{\cdot}{x}}_{3} = \frac{1}{L} (V - R x_{3} - n K_{g} x_{2})

(21)

where

\begin{matrix} A_{0} = \frac{1}{A (ϕ_{2}) + n^{2} J} \\ A_{1} = - \frac{1}{2} \frac{\partial A (ϕ_{2})}{\partial ϕ_{2}} \\ A_{2} = - n^{2} B \\ A_{3} = - n T_{L} \end{matrix}

Note that the control input to the system is now the voltage v and the output variable is the angular velocity of the crank ( $x_{2}$ ).

Velocity controller

The control scheme described in this work consists of two loops. The first one obtains the current reference ( $x_{3}^{*}$ ) that is a virtual control signal as a function of the velocity error. The second loop determines the voltage applied to the motor considering the current error. Linearization via feedback and PID controllers is applied in both the control loops to assure the correct velocity regulation in the coupler point.

The control synthesis begins defining the current tracking error as $e_{3} = x_{3} - x_{3}^{*}$ . Now taking into account the equation for the current in equation (20), the error dynamics ${\overset{\cdot}{e}}_{3}$ can be written as follows

{\overset{\cdot}{e}}_{3} = \frac{1}{L} (V - R x_{3} - n k_{g} x_{2}) - {\overset{\cdot}{x}}_{3}^{*}

(22)

From equation (22), a linearizing PID control signal can be proposed as

V = R x_{3} + n k_{g} x_{2} + L {\overset{\cdot}{x}}_{3}^{*} - L (k_{p} e_{3} + k_{i} η + k_{d} {\overset{\cdot}{e}}_{3})

(23)

where

\overset{\cdot}{η} = e_{3}

Introducing the control law (23) into the error dynamics (22), this is reduced to

\begin{matrix} {\overset{\cdot}{e}}_{3} = - k_{p} e_{3} - k_{i} η - k_{d} {\overset{\cdot}{e}}_{3} \\ = \frac{- k_{p} e_{3} - k_{i} η}{1 + k_{d}} \end{matrix}

(24)

where the error dynamics can be globally and asymptotically stable if the gains $k_{p}$ , $k_{i}$ , and $k_{d}$ are adequately selected.

As can be noted, if the control law (23) includes the current reference ( $x_{3}^{*}$ ) and its time derivative, the second control loop defines this virtual control variable. In order to determine this variable, a velocity error ( $e_{2} = x_{2} - x_{2}^{*}$ ) is defined, and considering the equation for the current in equation (20), the velocity error dynamics can be written as

{\overset{\cdot}{e}}_{2} = A_{0} (A_{1} x_{2}^{2} + A_{2} x_{2} + n K_{m} x_{3} + A_{3}) - {\overset{\cdot}{x}}_{2}^{*}

(25)

Replacing $x_{3}$ with $e_{3} + x_{3}^{*}$ , we can obtain

{\overset{\cdot}{e}}_{2} = A_{0} (A_{1} x_{2}^{2} + A_{2} x_{2} + n K_{m} (e_{3} + x_{3}^{*}) + A_{3}) - {\overset{\cdot}{x}}_{2}^{*}

(26)

Taking into account that the first control loop quickly assures $e_{3} \to 0$ , equation (26) is reduced to

{\overset{\cdot}{e}}_{2} = A_{0} (A_{1} x_{2}^{2} + A_{2} x_{2} + n K_{m} x_{3}^{*} + A_{3}) - {\overset{\cdot}{x}}_{2}^{*}

(27)

As $x_{3}^{*}$ is considered a virtual control signal, it can be proposed as follows

\begin{matrix} x_{3}^{*} = \frac{- 1}{n k_{m}} [A_{1} x_{2}^{2} + A_{2} x_{2} + A_{3} \\ + \frac{k_{p 2} e_{2} + k_{i 2} ξ + k_{d 2} {\overset{\cdot}{e}}_{2} - {\overset{\cdot}{x}}_{2}^{*}}{A_{0}}] \end{matrix}

(28)

where

\overset{\cdot}{ξ} = e_{2}

Then, replacing the control law (28) into the velocity error dynamics (27), we can obtain

\begin{matrix} {\overset{\cdot}{e}}_{2} = - k_{p 2} e_{2} - k_{i 2} ξ - k_{d 2} {\overset{\cdot}{e}}_{2} \\ = \frac{- k_{p 2} e_{2} - k_{i 2} ξ}{1 + k_{d 2}} \end{matrix}

(29)

In order to verify the stability of the complete closed-loop system, a Lyapunov candidate function is proposed as

W (z) = z^{T} Kz > 0

(30)

where

z = {[e_{2} e_{3} η ξ]}^{T}

and

K = [\begin{matrix} 1 + k_{d 2} & 0 & 1 + k_{d 2} & 0 \\ 0 & 1 + k_{d} & 0 & 1 + k_{d} \\ 1 + k_{d 2} & 0 & k_{p 2} + k_{i 2} & 0 \\ 0 & 1 + k_{d} & 0 & k_{p} + k_{i} \end{matrix}]

where $k_{p} + k_{i} > 1 + k_{d} > 0$ and $k_{p 2} + k_{i 2} > 1 + k_{d 2} > 0$ , guarantee that matrix K is positive definite.

Then, if $\overset{\cdot}{W} (z) < 0 \forall z \in ℜ - {0}$ , the global and asymptotical stability condition of the system is demonstrated. So, this derivative can be written as

\begin{matrix} \overset{\cdot}{W} (z) = {\overset{\cdot}{e}}^{T} Ke \\ = - (k_{p 2} - k_{d 2} - 1) e_{2}^{2} - (k_{p} - k_{d} - 1) e_{3}^{2} \\ - k_{i 2} ξ^{2} - k_{i} η^{2} \end{matrix}

The above result satisfies the stability condition that if $k_{p} > 1 + k_{d}$ , $k_{p} > 1 + k_{d}$ , $k_{i} > 0$ , and $k_{i 2} > 0$ , the origin of the error dynamics of the complete closed-loop system is the unique stability point and is global and asymptotically stable.

It is important to remark that in the stability proof, the error $e_{1} = x_{1} - x_{1}^{*}$ was not explicitly included because it is identical to the variable $ξ$ . The complete control scheme for the four-bar linkage can be observed in Figure 4.

Figure 4.

Complete control scheme.

Simulation results

In order to verify the performance of the proposed control law, simulations for two different four-bar linkages, each one with its own path, were conducted. The two mechanisms and their respective paths are presented in Figure 5. For convenience, from now on, they are appointed as mechanism 1 and mechanism 2. The test consists of regulating the coupler point linear velocity $V_{P}$ of the mechanism at $0.3 m / s$ through tracking an angular velocity $(ω_{2})$ profile in the crank. The mechanisms and motor parameters are reported in Tables 1 and 2, respectively. The controller gains are presented in Table 3.

Figure 5.

Four-bar mechanisms with coupler point paths: (a) mechanism 1 and (b) mechanism 2.

Table 1.

Mechanism parameters.

Parameter	Value (mechanism 1)	Value (mechanism 2)
$L_{1} (m)$	0.10891	0.3972
$L_{2} (m)$	0.042259	0.0588
$L_{3} (m)$	0.09644	0.2351
$L_{4} (m)$	0.058781	0.22716
$r_{cx} (m)$	0.039137	0.403779
$r_{cy} (m)$	0.04295	0.093921
$J_{2} (kg \cdot m^{2})$	$1.51 \times 10^{- 5}$	$2.76 \times 10^{- 5}$
$J_{3} (kg \cdot m^{2})$	$3.5468 \times 10^{- 3}$	$3.5468 \times 10^{- 3}$
$J_{4} (kg \cdot m^{2})$	$1.7623 \times 10^{- 6}$	$3.8779 \times 10^{- 4}$
$m_{2} (kg)$	0.03349	0.04234
$m_{3} (kg)$	0.2586	0.2586
$m_{4} (kg)$	0.00547	0.08156
$α (rad)$	0.32195	5.83047

Table 2.

Motor parameters.

Parameter	Value
R ( $Ω$ )	2
L (H)	1
$K_{m} (N \cdot m / A)$	0.260
$K_{g} (v \cdot s)$	0.260
$J (Kg \cdot m^{2})$	0.011
$T_{L} (N \cdot m)$	0.28
$B (N \cdot m \cdot s)$	0

Table 3.

Controller gains.

Parameter	Value
$K_{p}$	3000
$K_{d}$	200
$K_{i}$	50
$K_{p 2}$	10.8
$K_{d 2}$	0
$K_{i 2}$	100

Mechanism 1 simulation

The path generated by the coupler point mechanism 1 is presented in Figure 6(a). So as to ensure that the coupler point travels its path at desired speed, the Crank of the mechanism must follow the angular velocity profile shown in Figure 6(b). In Figure 7(a), it can be observed that maximum error in the profile tracking is about 0.33 rad/s. Consequently, the obtained linear velocity of the coupler point error is over 0.017 m/s. Figure 7(b) evidences this fact. Figure 7(c) and (d) present both voltage and current requirements needed to complete the task. It can be noted that the magnitude of this variables is bounded, so the control is realizable, because power consumption is finite. As can be observed, the imposed speed for the path of mechanism 1 is reached without drawbacks. So, a test under more severe conditions is necessary to verify the performance of the proposed control scheme; by this motive, mechanism 2 was selected. Where mechanical architecture is of the same type, with slightly different dimensions and inertial properties, the path generated to a point of the coupler link has an abrupt change in direction in an instant of time very small.

Figure 6.

(a) Mechanism 1 path and (b) $ω_{2}$ profile for V_P = 0.3 m/s.

Figure 7.

Mechanism 1 behavior: (a) $ω_{2}$ tracking, (b) V(_P) regulation, (c) motor voltage, and (d) motor current.

Mechanism 2 simulation

In this case, the path of the coupler point of mechanism 2 is shown in Figure 8(a), and the angular velocity profile is shown in Figure 8(b). The profile tracking is described in Figure 9(a), where the maximum error is about 0.39 rad/s. Thus, the linear velocity of the coupler point error is over 0.013 m/s. This result verifies that the control objective satisfactorily complies. However, it can be seen that there are voltage peaks and current peaks at 180° of the crank, corresponding to the inflection point of the trajectory, as it is exhibited in Figure 9(c) and (d). Despite these peaks, it is important to mention that the feasibility of using the proposed control is because the peak voltage and current values are bounded.

Figure 8.

(a) Mechanism 2 path and (b) $ω_{2}$ profile for V(P_c) = 0.3 m/s.

Figure 9.

Mechanism 2 behavior: (a) $ω_{2}$ tracking, (b) V(P_c) regulation, (c) motor voltage, and (d) motor current.

Conclusion

The PID controller used in this work for the motor-mechanism coupling, with the target to control the linear velocity coupler point ( $V_{p}$ ), was able to fulfill the described task. Despite $V_{p}$ was not directly measured, PID controller satisfies this task properly. It is important to mention that the path profile followed by the coupler point is not smooth in all its trajectories; however, it is observed that the simulation results for $V_{p}$ error has a maximum of $0.0013 m / s$ for $V_{p}$ desired of $0.3 m / s$ . The examples considered illustrate the effectiveness of the employed methodology.

Footnotes

Acknowledgements

The authors thank PRODEP, Conacyt, and SNI for their support.

Academic Editor: Yangmin Li

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: They received financial support by part of PRODEP for publication of this article.

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