Research on high precision motion control method of automated production line based on adaptive control

1)

Industrial Camera

In this paper, a CMOS industrial camera is used to collect static object images. Experimental visual inspection object is self-designed irregular parts, horizontal size of 65mm, vertical size of 30mm, tolerance requirements of 0.2mm, the edge of the software repeatability empirical value of 0.7, the camera selection process to calculate the parameters are as follows:

Pixel equivalent: Res = 0.02/0.7 = 0.02857mm/pixel

Short side vision: H = 30 × 1.5 = 45mm

Long side view: W = 45/(3/4) = 60mm

Short-edge resolution: H_lr = 45/0.02857 = 1575.02362 pixel

Long-edge resolution: W_tr = 60/0.02857 = 2100.03150 pixel

1)

Coordinate system and transformation relationship

Assume that the physical dimensions of a pixel (μ, v) in the pixel coordinate system in the direction of x and y axes of the image plane are dx and dy respectively, and that the μ and v axes are perpendicular to each other. The coordinates of this pixel in the image coordinate system are expressed as follows: (1){ μ=xdx+μ0 v=ydy+v0$$\left\{ {\begin{array}{*{20}{l}} {\mu = \frac{x}{{dx}} + {\mu _0}} \\ {v = \frac{y}{{dy}} + {v_0}} \end{array}} \right.$$

For the convenience of subsequent coordinate transformation calculations, Eq. (1) is rewritten as: (2)[ μ v 1]=[ 1dx 0 μ0 0 1dy v0 0 0 1]⋅[ x y 1]$$\left[ {\begin{array}{*{20}{c}} \mu \\ v \\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{dx}}}&0&{{\mu _0}} \\ 0&{\frac{1}{{dy}}}&{{v_0}} \\ 0&0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{l}} x \\ y \\ 1 \end{array}} \right]$$

Due to the uneven placement of the object, lens distortion [20] and other factors will lead to μ axis and v axis are not perpendicular to each other, θ is the angle between the two axes, then in this case the image coordinate system and pixel coordinate system in the point of the coordinate transformation relationship is as follows: (3){ μ=xdx−ycotθdx+μ0 v=ydvsinθ+v0$$\left\{ {\begin{array}{*{20}{c}} {\mu = \frac{x}{{dx}} - \frac{{y\cot \theta }}{{dx}} + {\mu _0}} \\ {v = \frac{y}{{dv\sin \theta }} + {v_0}} \end{array}} \right.$$

Rewriting equation (3) into matrix form yields: (4)[ μ v 1]| 1dx −1dxcotθ μij 0 1dysinθ ti 0 0 1|⋅[ x y 1]$$\left[ {\begin{array}{*{20}{c}} \mu \\ v \\ 1 \end{array}} \right]\left| {\begin{array}{*{20}{c}} {\frac{1}{{dx}}}&{ - \frac{1}{{dx}}\cot \theta }&{{\mu _{ij}}} \\ 0&{\frac{1}{{dy\sin \theta }}}&{{t_i}} \\ 0&0&1 \end{array}} \right| \cdot \left[ {\begin{array}{*{20}{l}} x \\ y \\ 1 \end{array}} \right]$$

A point (x,y)$$\left( {x,y} \right)$$ in the image plane can be transformed from the image coordinate system to the camera coordinate system through equation (5), and this transformation relationship can be derived from the phase center principle of the imaging of the pinhole model: (5){ x=fxZc y=fZc$$\left\{ {\begin{array}{*{20}{l}} {x = \frac{{fx}}{{{Z_c}}}} \\ {y = \frac{f}{{{Z_c}}}} \end{array}} \right.$$

Eq. (5) where f is the focal length, rewrite Eq. (5) in matrix form: (6)Zc⋅[ x y 1]=[ f 0 0 0 0 f 0 0 0 0 f 1]⋅[ Xc Yc Zc 1]$${Z_c} \cdot \left[ {\begin{array}{*{20}{c}} x \\ y \\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} f&0&0&0 \\ 0&f&0&0 \\ 0&0&f&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{X_c}} \\ {{Y_c}} \\ {{Z_c}} \\ 1 \end{array}} \right]$$

Transforming a point from the world coordinate system to the camera coordinate system belongs to the rigid transformation [21], this rigid transformation consists of translation vectors and rotation matrices, the transformation relationship between the world coordinate system and the camera coordinate system is: (7)[ Xc Yc Zc 1]=[ R3×3 T3×1 0 1]⋅[ Xw Yw Zw 1]=M2⋅[ Xv Yw Zw 1]$$\left[ {\begin{array}{*{20}{c}} {{X_c}} \\ {{Y_c}} \\ {{Z_c}} \\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{R_{3 \times 3}}}&{{T_{3 \times 1}}} \\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{X_w}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right] = {M_2} \cdot \left[ {\begin{array}{*{20}{c}} {{X_v}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right]$$

In Eq. (7), R_3×3 = R(α, β, γ) is a rotation matrix, and in R_3×3, α, β, and γ are the rotation angles around the x, y, and z axes of the camera coordinate system, respectively; T3×1=(tx,ty,tz)T$${T_{3 \times 1}} = {\left( {{t_x},{t_y},{t_z}} \right)^T}$$ is a 3D translation vector; M₂ is a 4 × 4 matrix, and α, β, γ, and t_x, t_y, and t_z are the six parameters that constitute the external parameters of the camera.

Equation (8) in f, s_x, s_y, μ₀ and v₉ belong to the internal parameters of the camera, which can be used to determine the transformation relationship between the world coordinate system and the image coordinate system in the camera coordinate system, and the internal and external parameters of the industrial camera can be determined by the industrial camera calibration. (8)Zc⋅[ μ v 1]=[ 1dx −1dxcotθ μ0 0 1dysinθ v0 0 0 1]⋅[ f 0 0 0 0 f 0 0 0 0 f 1 ]⋅[ R3×3 T3x1 0 1]⋅[ Xw Yw Zw 1] =[ fdx −fdxcotθ μ0f μ0 0 fdysinθ v0f v0 0 0 f 1]⋅[ R3×3 T3×1 0 1]⋅[ Xw Yw Zw 1] =[ sz −sxcotθ μ0f μ0 0 sysinθ v0f v0 0 0 f 1]⋅[ R333 T3×1 0 1]⋅[ Xw Yw Zw 1] =M1M2⋅[ Xw Yw Zw 1]$$\begin{array}{rcl} {Z_c} \cdot \left[ {\begin{array}{*{20}{c}} \mu \\ v \\ 1 \end{array}} \right] &=& \left[ {\begin{array}{*{20}{c}} {\frac{1}{{dx}}}&{ - \frac{1}{{dx}}\cot \theta }&{{\mu _0}} \\ 0&{\frac{1}{{dy\sin \theta }}}&{{v_0}} \\ 0&0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} f&0&0 \\ 0&{}&{} \\ 0&f&0 \\ 0&{}&{} \\ 0&0&f \\ 1&{}&{} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{R_{3 \times 3}}}&{{T_{3x1}}} \\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{X_w}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right] \\ &=& \left[ {\begin{array}{*{20}{c}} {\frac{f}{{dx}}}&{ - \frac{f}{{dx}}\cot \theta }&{{\mu _0}f}&{{\mu _0}} \\ 0&{\frac{f}{{dy\sin \theta }}}&{{v_0}f}&{{v_0}} \\ 0&0&f&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{R_{3 \times 3}}}&{{T_{3 \times 1}}} \\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{X_w}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right] \\ &=& \left[ {\begin{array}{*{20}{c}} {{s_z}}&{ - {s_x}\cot \theta }&{{\mu _0}f}&{{\mu _0}} \\ 0&{\frac{{{s_y}}}{{\sin \theta }}}&{{v_0}f}&{{v_0}} \\ 0&0&f&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{R_{333}}}&{{T_{3 \times 1}}} \\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {{X_w}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right] \\ &=& {M_1}{M_2} \cdot \left[ {\begin{array}{*{20}{c}} {{X_w}} \\ {{Y_w}} \\ {{Z_w}} \\ 1 \end{array}} \right] \\ \end{array}$$ 2)

Camera imaging model

In practice, due to the optical industrial lenses in the lens processing and manufacturing precision and camera assembly process there are errors, resulting in the projection of points in three-dimensional space to the image coordinate system is not the position of the aberration of the position of a slight offset.

Generally optical lens tangential aberration is small, where p₁ and p₂ are radial aberration coefficients. The amount of specific correction can be expressed in equation (9): (9){ σwi=2p1x+p1(r2+2x2) σwi=2p2x+p1(r2+2y2) r2=(x2−μwi)+(y2−vn)2$$\left\{ {\begin{array}{*{20}{l}} {{\sigma _{wi}} = 2{p_1}x + {p_1}\left( {{r^2} + 2{x^2}} \right)} \\ {{\sigma _{wi}} = 2{p_2}x + {p_1}\left( {{r^2} + 2{y^2}} \right)} \\ {{r^2} = \left( {{x^2} - {\mu _{wi}}} \right) + {{\left( {{y^2} - {v_n}} \right)}^2}} \end{array}} \right.$$

k₁, k₂, k₃, p₁ and p₂ are also internal camera parameters. The transformation relationship between the ideal coordinates and the actual coordinates is shown below: (10){ x=x′+(x′−μ0)(k1r2+k2r4+k3r6)+2p1xy+p2(r2+2x2) y=y′+(y′−v0)(k1r2+k2r4+k3r6)+2p2xy+p1(r2+2y2)$$\left\{ {\begin{array}{*{20}{l}} {x = x\prime + \left( {x\prime - {\mu _0}} \right)\left( {{k_1}{r^2} + {k_2}{r^4} + {k_3}{r^6}} \right) + 2{p_1}xy + {p_2}\left( {{r^2} + 2{x^2}} \right)} \\ {y = y\prime + \left( {y\prime - {v_0}} \right)\left( {{k_1}{r^2} + {k_2}{r^4} + {k_3}{r^6}} \right) + 2{p_2}xy + {p_1}\left( {{r^2} + 2{y^2}} \right)} \end{array}} \right.$$ 3)

HALCON-based industrial camera calibration

In this paper, the calibration model is created under the conditions of the HALCON vision algorithm [22] library; the images of calibration plates with different placement positions are acquired; the industrial camera is calibrated, and the internal and external parameters of the camera are acquired to correct the images. The final parameter that will be solved for the industrial robot in the EIH system is the chi-square transformation matrix of the camera coordinate system with respect to the robot end coordinate system.

In the EIH system, the following system of equations holds for two arbitrary bit positions during robot movement: (11){ ewHi*ceH=gwH*cgHi ewHj*ceH=gwH*cgHj$$\left\{ {\begin{array}{*{20}{l}} {_e^w{H_i}*_c^eH = _g^wH*_c^g{H_i}} \\ {_e^w{H_j}*_c^eH = _g^wH*_c^g{H_j}} \end{array}} \right.$$

Multiply both sides of the equal sign of the equation above the agenda group by (cgHi)−1$${\left( {_c^g{H_i}} \right)^{ - 1}}$$, and both sides of the equal sign of the equation below the agenda group by (cgHj)−1$${\left( {_c^g{H_j}} \right)^{ - 1}}$$: (12){ ewHi*ceH*(cgHi)−1=gwH ewHj*ceH*(cgHj)−1=gwH$$\left\{ {\begin{array}{*{20}{l}} {_e^w{H_i}*_c^eH*{{\left( {_c^g{H_i}} \right)}^{ - 1}} = _g^wH} \\ {_e^w{H_j}*_c^eH*{{\left( {_c^g{H_j}} \right)}^{ - 1}} = _g^wH} \end{array}} \right.$$

Combining the two equations above and below the agenda group and eliminating the constant matrix gwH$$_g^wH$$ yields: (13)ewHi*ceH*(cgHi)−1=ewHi*ceH*(cgHi)−1$$_e^w{H_i}*_c^eH*{\left( {_c^g{H_i}} \right)^{ - 1}} = _e^w{H_i}*_c^eH*{\left( {_c^g{H_i}} \right)^{ - 1}}$$

The above equation is obtained by multiplying both sides of the equal sign simultaneously left and right: (14)(cwHj)−1*cwHi*ceH=ceH*(cgHj)−1*(tgHt)$${\left( {_c^w{H_j}} \right)^{ - 1}}*_c^w{H_i}*_c^eH = _c^eH*{\left( {_c^g{H_j}} \right)^{ - 1}}*\left( {_t^g{H_t}} \right)$$

Substitute (egHj)−1=(geHj)$${\left( {_e^g{H_j}} \right)^{ - 1}} = \left( {_g^e{H_j}} \right)$$ and (egHi)=(geHi)−1$$\left( {_e^g{H_i}} \right) = {\left( {_g^e{H_i}} \right)^{ - 1}}$$ into the above equation to obtain: (15)(ewHj)−1*ewHj*ceHi=ceH*(geHj)*(geHj)−1$${\left( {_e^w{H_j}} \right)^{ - 1}}*_e^w{H_j}*_c^e{H_i} = _c^eH*\left( {_g^e{H_j}} \right)*{\left( {_g^e{H_j}} \right)^{ - 1}}$$

Let X=ceH$$X = _c^eH$$, A=(ewHj)−1*cwHj$$A = {\left( {_e^w{H_j}} \right)^{ - 1}}*_c^w{H_j}$$, B=(gcHj)*(geHi)−1$$B = \left( {_g^c{H_j}} \right)*{\left( {_g^e{H_i}} \right)^{ - 1}}$$, you can simplify the above equation to AX = XB form.

In industrial applications usually use the nine-point calibration method to determine the camera coordinate system and mechanical end coordinates of each household transformation relationship, through the camera to obtain nine Mark points in the camera coordinate system under the image coordinates, control the robot end of the actuator in the actuator aligned to the nine Mark points in turn, to obtain the coordinates of the Mark points in the robot coordinate system, the use of the nine sets of corresponding coordinates can be set up as shown in the formula (16) Using the nine sets of corresponding coordinates, a coordinate transformation matrix can be established as shown in equation (16), where R is the rotation matrix and T is the translation matrix. (16)HomMat2D=[ R ⋯ T ⋮ ⋱ ⋮ 0 ⋯ 1]=H(T)⋅H(R)$$HomMat2D = \left[ {\begin{array}{*{20}{c}} R& \cdots &T \\ \vdots & \ddots & \vdots \\ 0& \cdots &1 \end{array}} \right] = H(T) \cdot H(R)$$

Based on the fact that AGVs can accomplish omnidirectional movements during operation. Understanding the principle of kinematics can better control the omnidirectional movement of AGV. Model the forward and reverse kinematics of an AGV based on a McNamee wheel. Establish a coordinate system for the kinematic model of the AGV fitted with a McNamee wheel. Take the positive center of the AGV as the coordinate origin, the X-axis positive horizontal to the left, the Y-axis positive vertical upward, and the ω-rotation positive counterclockwise. Let the translational velocity of the AGV be v, the orthogonal decomposition velocities be v_x and v_y, and the rotational angular velocity be ω, which is expressed by Eq: (17){ v→=vx→+vy→ ω$$\left\{ {\begin{array}{*{20}{l}} {\vec v = \overrightarrow {{v_x}} + \overrightarrow {{v_y}} } \\ \omega \end{array}} \right.$$

The four wheels of the AGV are labeled, clockwise from the upper left wheel as wheel 1, 2, 3 and 4. Taking wheel 1 as an example, the speed of the wheel axis is v₁, and v₁ is synthesized by the translational speed v and rotational speed v_ω1, i.e.: (18){ v1→=v→+vω1→ vω1→=ω→×l→ l→=a→+b→$$\left\{ {\begin{array}{*{20}{l}} {\overrightarrow {{v_1}} = \vec v + \overrightarrow {{v_{\omega 1}}} } \\ {\overrightarrow {{v_{\omega 1}}} = \vec \omega \times \vec l} \\ {\vec l = \vec a + \vec b} \end{array}} \right.$$

v₁ is decomposed orthogonally into v_1x and v_1y, and from Eq. (17) and Eq. (12) the relation with the given speeds v_x and v_y can be introduced as: (19){ v1→=v1x→+vty→ v1x=vx+ω⋅b v1y=vy−ω⋅a$$\left\{ {\begin{array}{*{20}{l}} {\overrightarrow {{v_1}} = \overrightarrow {{v_{1x}}} + \overrightarrow {{v_{ty}}} } \\ {{v_{1x}} = {v_x} + \omega \cdot b} \\ {{v_{1y}} = {v_y} - \omega \cdot a} \end{array}} \right.$$

The kinematic model of the roller in the wheel where it meets the ground, the speed of the roller is consistent with the wheel’s axial speed as v_l, the speed of the roller can be decomposed into v_∥ and v_⊥, v_∥ is the speed component along the axis of the roller, v_⊥ is the speed component in the direction perpendicular to the axis of the roller, which can be obtained from the knowledge of vectors: (20){ vli=v→1⋅u→ vli=−22v1z+22v1y$$\left\{ {\begin{array}{*{20}{l}} {{v_{li}} = {{\vec v}_1} \cdot \vec u} \\ {{v_{li}} = - \frac{{\sqrt 2 }}{2}{v_{1z}} + \frac{{\sqrt 2 }}{2}{v_{1y}}} \end{array}} \right.$$

u→$$\vec u$$ is the unit vector in direction v_∥.

v_r1 is the linear velocity of the wheel, and the angle between v_r and v_∥ is 45°, which can be deduced from equation (20): (21)vr1=vijcos45°=v1y−v1x=vy−vx−ω(a+b)$${v_{r1}} = \frac{{{v_{ij}}}}{{\cos 45^\circ }} = {v_{1y}} - {v_{1x}} = {v_y} - {v_x} - \omega (a + b)$$

Similarly, the linear velocities v_r2, v_r3, and v_r4 of wheels 2, 3, and 4 can be obtained, thus obtaining the inverse kinematic equations of the AGV: (22){ vr1=v1y−v1z=vy−vx−ω(a+b) vr2=v2x+v2y=vy+vx+ω(a+b) vr3=v3y−v3x=vy−vx+ω(a+b) vr4=v4x+v4y=vy+vx−ω(a+b)$$\left\{ {\begin{array}{*{20}{l}} {{v_{r1}} = {v_{1y}} - {v_{1z}} = {v_y} - {v_x} - \omega (a + b)} \\ {{v_{r2}} = {v_{2x}} + {v_{2y}} = {v_y} + {v_x} + \omega (a + b)} \\ {{v_{r3}} = {v_{3y}} - {v_{3x}} = {v_y} - {v_x} + \omega (a + b)} \\ {{v_{r4}} = {v_{4x}} + {v_{4y}} = {v_y} + {v_x} - \omega (a + b)} \end{array}} \right.$$

Let the radius of the wheel be R, v_ir = ω_iR brought into the above equation and written in matrix form: (23)[ ω1 ω2 ω3 ω4]R=[ 1 −1 −(a+b) 1 1 a+b 1 −1 a+b 1 1 −(a+b)][ vy vx ω]$$\left[ \begin{array}{rcl} {\omega _1} \\ {\omega _2} \\ {\omega _3} \\ {\omega _4} \\ \end{array} \right]R = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&{ - (a + b)} \\ 1&1&{a + b} \\ 1&{ - 1}&{a + b} \\ 1&1&{ - (a + b)} \end{array}} \right]\left[ \begin{array}{rcl} {v_y} \\ {v_x} \\ \omega \\ \end{array} \right]$$

The positive kinematic equation of the AGV is obtained from the above equation: (24){ vx=R4(−ω1+ω2−ω3+ω4) vy=R4(ω1+ω2+ω3+ω4) ω=R4(a+b)(−ω1+ω2+ω3−ω4)$$\left\{ {\begin{array}{*{20}{l}} {{v_x} = \frac{R}{4}\left( { - {\omega _1} + {\omega _2} - {\omega _3} + {\omega _4}} \right)} \\ {{v_y} = \frac{R}{4}\left( {{\omega _1} + {\omega _2} + {\omega _3} + {\omega _4}} \right)} \\ {\omega = \frac{R}{{4(a + b)}}\left( { - {\omega _1} + {\omega _2} + {\omega _3} - {\omega _4}} \right)} \end{array}} \right.$$

The AGV is controlled by the forward and inverse kinematics equations of the AGV, which are given (vx,vy,ω)$$\left( {{v_x},{v_y},\omega } \right)$$ by the main control system, and the rotational speeds of the four wheels (ω1,ω2,ω3,ω4)$$\left( {{\omega _1},{\omega _2},{\omega _3},{\omega _4}} \right)$$ are calculated by equation (23), and the Robo Module DC servo motor driver will control the motors to output the corresponding rotational speeds according to the corresponding rotational speed parameters.

The key to AGV motion control [23] is the control of the motor speed, DC motors for speed regulation generally use PWM speed regulation, i.e., pulse width modulation, to control the motor speed by changing the ratio of the motor armature voltage on-time to the energization time. In this paper, this method is used to regulate the speed of the motor. Figure 1 shows the AGV action control flow.

Figure 1.

AGV action control process

When the AGV is running in a straight line, the same PWM value is used to control the four motors to output the same rotational speed, and the AGV travels at a constant speed. When AGV performs position adjustment, or performs turning action or panning action at special intersections, the master control system outputs different PWM adjustment parameters to the motors, and adjusts the rotational speed and steering of each motor to complete the corresponding action. When the AGV body moves away from the navigation line, the vision system determines the deviation between the navigation line and the center of the body. The diagonal line in the figure is the line center line extracted by the vision system, and the angle between the line center line and Y axis is θ, which indicates the angle offset between the AGV and the route, and the larger the value of θ, the larger the offset is; a is the intercept of the line center line on X axis, which indicates the distance deviation between the line center and the midpoint of the field of view, and the larger the value of a, the larger the distance of the AGV deviation from the route is. At this time, it is necessary for the AGV control system to output the PWM adjustment parameters of the corresponding McNamee wheel motor according to the size of the deviation, change the motor speed, and utilize the rotational speed difference between the wheels to complete the position adjustment.

Conventional PID control [24] is a linear controller whose control is a typical unit negative feedback control.

The input quantity is the deviation e(t) between the set value r(t) and the output value y(t), and the control quantity u(t) is formed by using a linear combination of proportional, integral, and differential, which stabilizes the output at the desired value by adjusting the controlled object. The role of the proportional link stage is used to eliminate the error, affecting the dynamic response speed and the ability to eliminate the error; the role of the integral link is to eliminate the steady state error generated by the proportional link, to improve the system’s degree of non-differential; the role of the differential link is to introduce correction signals in advance, suppressing the system’s overshooting and oscillations, and to improve the stability of the system.

Conventional PID control of the control object when the control law can be expressed as: (25)u(t)=Kp[e(t)+1Ti∫0te(t)dt+Tdde(t)dt]$$u(t) = {K_p}\left[ {e(t) + \frac{1}{{{T_i}}}\int_0^t e (t)dt + {T_d}\frac{{de(t)}}{{dt}}} \right]$$

Where K_p is the proportionality coefficient, T_i is the integral time constant, T_d is the differential time constant.

After the further development of microcontroller chips and computers, the digital PID to complete the program to slowly replace the analog PID control circuit. Through the program to achieve the PID control not only to complete the control task, and control is more versatile.PID control law equation is continuous, due to the digital PID control is a sampling control, can only be sampling the moment of deviation value to the operation, so the need for PID control law discretization process.

The following approximation can be made before discretization: (26)u(t)≈u(k)$$u(t) \approx u(k)$$ (27)e(t)≈e(k)$$e(t) \approx e(k)$$ (28)∫0te(t)dt=∑j=0ke(j)Δt=∑j=0kTe(j)$$\int_0^t e (t){\text{d}}t = \sum\limits_{j = 0}^k e (j)\Delta t = \sum\limits_{j = 0}^k T e(j)$$ (29)de(t)dt≈e(k)−e(k−1)Δt=e(k)−e(k−1)T$$\frac{{de(t)}}{{dt}} \approx \frac{{e(k) - e(k - 1)}}{{\Delta t}} = \frac{{e(k) - e(k - 1)}}{T}$$

Eq. (28), Eq. (29) where T is the sampling period and k is the sampling sequence number (k = 0, 1, 2, ⋯). Two forms of PID algorithms, positional PID and incremental PID, can be obtained using this approximation.

The discretized positional PID control law can be expressed as: (30)u(k) = Kp{e(k)+TTi∑j=0ke(j)+TdT[e(k)−e(k−1)]} = Kpe(k)+Ki∑j=0ke(j)+Kd[e(k)−e(k−1)]$$\begin{array}{rcl} u(k) &=& {K_p}\left\{ {e(k) + \frac{T}{{{T_i}}}\sum\limits_{j = 0}^k e (j) + \frac{{{T_d}}}{T}[e(k) - e(k - 1)]} \right\} \\ &=& {K_p}e(k) + {K_i}\sum\limits_{j = 0}^k e (j) + {K_d}[e(k) - e(k - 1)] \\ \end{array}$$

From equation (30), it can be seen that the control quantity u(k) of the positional PID output is related to all the past deviation signals, and when the integral term continues to accumulate the error and reaches saturation, it will still continue to integrate the error. However, once the direction of the error is reversed, the system has to have a certain process to exit the saturation state. Since the output of each control quantity is related to the past state, it increases the workload of operation, and the system is prone to drastic changes once there is a problem with the control state. In order to reduce the amount of arithmetic, reduce the impact of false action, it can be rewritten as an incremental PID.

The control volume output of the controller at k − 1 sampling moments is: (31)u(k−1) = Kp{e(k−1)+TTi∑j=0k−1e(j)+TdT[e(k−1)−e(k−2)]} = Kpe(k−1)+Ki∑j=0k−1e(j)+Kd[e(k−1)−e(k−2)]$$\begin{array}{rcl} u(k - 1) &=& {K_p}\left\{ {e(k - 1) + \frac{T}{{{T_i}}}\sum\limits_{j = 0}^{k - 1} e (j) + \frac{{{T_d}}}{T}[e(k - 1) - e(k - 2)]} \right\} \\ &=& {K_p}e(k - 1) + {K_i}\sum\limits_{j = 0}^{k - 1} e (j) + {K_d}[e(k - 1) - e(k - 2)] \\ \end{array}$$

Then the incremental PID control equation is: (32)Δu(k)=u(k)−u(k−1) =Kp{[e(k)−e(k−1)]+TTie(k)+TdT[e(k)−2e(k−1)+e(k−2)]}$$\begin{array}{l} \Delta u(k) = u(k) - u(k - 1) \\ = {K_p}\left\{ {[e(k) - e(k - 1)] + \frac{T}{{{T_i}}}e(k) + \frac{{{T_d}}}{T}[e(k) - 2e(k - 1) + e(k - 2)]} \right\} \\ \end{array}$$

In the above equation K_p is the proportional coefficient, Ki=KpTTi$${K_i} = {K_p}\frac{T}{{{T_i}}}$$ is the integral coefficient and Kd=KpTdT$${K_d} = {K_p}\frac{{{T_d}}}{T}$$ is the differential coefficient. From the above equation, it can be seen that the control quantity Δu(k) is only related to the sampling value of the last three times, and there is no accumulation of the error of the past state, and the calculation error has less influence on the output of the control quantity, so that it can be weighted to get a better control effect, and it will not have a serious impact on the system work when there is a problem with the control of the system.

Fuzzy control is a control method based on fuzzy logic, and its main principle is to mimic the human way of thinking and transform the fuzzy and uncertain information into control actions.

The fuzzy controller is mainly composed of fuzzification interface, knowledge base, fuzzy inference and defuzzification interface, and the fuzzy controller composition is shown in Figure 2.

Figure 2.

Fuzzy controller components

There are various defuzzification methods such as maximum affiliation method, weighted average method, center of gravity method, etc. In this paper, the defuzzification method chosen is the center of gravity method, the specific formula is as follows: (33)u=∫abxμC(x)dx∫abμC(x)dx$$u = \frac{{\int_a^b x \mu C(x)dx}}{{\int_a^b \mu C(x)dx}}$$

In the above equation, u is the specific value after defuzzification, x is the element in the output domain, μC(x) is the affiliation function of the fuzzy set of the output quantity on the output domain, and [a, b] is the range of the output domain.

In order to ensure the control characteristics and accuracy in the motion of the automated production line, adaptive fuzzy PID control is used on the basis of using PID control. The block diagram of the adaptive fuzzy PID control system is shown in Figure 3. Adaptive fuzzy PID control combines the ideas of fuzzy control and adaptive control on the basis of the conventional PID algorithm, which is mainly used to improve the stability and robustness of the system compared to the conventional PID algorithm, and is suitable for complex time-varying nonlinear systems. Without changing the control effect of the original PID algorithm, the adaptive fuzzy PID controller calculates the control output through the deviation of turbidity and the rate of change of the deviation, and it can adaptively adjust the parameters of the PID controller in real time according to the actual operation of the system in order to maintain the stability and performance of the system, and to realize accurate control in the dynamically changing environment.

Figure 3.

Block diagram of adaptive fuzzy PID control system

Adaptive fuzzy PID control incrementally adjusts the three parameters of the PID controller through the fuzzy control rules established in the fuzzy controller. According to the fuzzy relationship between the deviation e of the set turbidity from the measured turbidity and the rate of change of the deviation ec and the fuzzy amount of the increment of the three parameters of the PID in the fuzzy controller, the turbidity deviation e and the rate of change of the deviation ec are updated in real time during the operation of the adaptive fuzzy PID control system, and the increment of the three parameters of the PID is rectified on-line according to the fuzzy control rules set up. The equations of K_p, K_i and K_d of the adaptive tuning PID controller are: (34)Kp=Kp0+λp⋅ΔKp$${K_p} = {K_{p0}} + {\lambda _p} \cdot \Delta {K_p}$$ (35)Ki=Ki0+λi⋅ΔKi$${K_i} = {K_{i0}} + {\lambda _i} \cdot \Delta {K_i}$$ (36)Kd=Kd0+λd⋅ΔKd$${K_d} = {K_{d0}} + {\lambda _d} \cdot \Delta {K_d}$$

Where K_p0, K_i0, K_d0 for the PID controller pre-adjusted initial parameters, ΔK_p, ΔK_i, ΔK_d after the fuzzy controller output corresponds to the fuzzy amount of K_p, K_i, K_d changes in the fuzzy amount of λ_p, λ_i, λ_d for ΔK_p, ΔK_i, ΔK_d, respectively, the proportionality factor. In the set turbidity and measured turbidity deviation e and deviation rate of change ec change, ΔK_p, ΔK_i, ΔK_d corresponding to the defuzzified value will also change. Therefore, K_p, K_i, K_d of the PID controller will adaptively adjust the parameters of the PID controller according to the real-time operating state of the control system to adapt to the dynamic changes in the system and external interference, which improves the adaptive ability of the control system, thus making it have a better control effect in all control processes.

The robot motion accuracy test results are shown in Fig. 7, with (a) and (b) representing the X-direction motion deviation and Y-direction motion deviation, respectively. The larger value of the motion deviation along the X-direction and Y-direction in different travel tests is taken as the motion deviation in that direction, and it can be seen from Fig. 7(a) that when the X-axis travel is 120mm in the negative direction, the magnitude of the motion deviation is the largest, which is between 0.002mm~0.035mm, and the motion deviation in the X-direction can be obtained to be ±0.016mm.From Fig. 7(b), when the Y-axis travel is 60mm in the positive direction, the magnitude of the motion deviation is the largest, which is between -0.29mm~0.035mm. 60mm, the amplitude of motion deviation is the largest, between -0.29mm~0.004mm, from which the motion deviation in Y direction can be obtained as ±0.017mm.

Figure 7.

Results of the robot motion accuracy test

The vision-guided right-angle coordinate-based robot trajectory planning algorithm proposed in this design can be oriented to arbitrary curves, obtain the discrete points of the target trajectory contour from the vision image, fit the motion trajectory by the arc-length parameterized Akim method, and guide the robot to realize the motion of any given trajectory by using the improved five-segment S-type velocity planning, which effectively improves the quality of the robot’s motion trajectory.

The visually acquired contour discrete points of any given trajectory are used as experimental data points, and the arc-length parameterized Akima trajectory fitting simulation test is carried out in Matlab 2020 environment, and 216 contour discrete points of the visually acquired trajectory are screened by the double chord error and chord tangent error constraints test, and the final selection of,50 data points is made to fit the Akima curves through the adjustment of error constraints, the results are shown in the following figure. The Akima curve simulation results are shown in Fig. 8, (a) is the Akima fitted curve, while (b)~(c) are the local zoomed-in plots of 1 and 2 in the fitted curve, respectively. The simulation data show that the obtained Akima curves can well depict the shape and convexity of the visual data points, and the maximum error between the fitted points and the theoretical points at the same X-coordinate is 0.114 mm, and the average error is 0.0287 mm.

Figure 8.

The Akima curve simulation results

After simulation and analysis, the continuous trajectory motion of a Cartesian Coordinate robot is applied to the gluing and sealing station of an assembly line. In order to obtain higher gluing quality, the motion speed of the glue gun needs to be limited, i.e., the path length and motion speed constraints are introduced in the velocity planning. The Akim curve interpolation path length obtained by the modified chord length parameter method is 1250 units, and the end actuator feed speed is selected as 1350 units/s. Four sets of experimental parameters, 3200, 1400, 5500, and 5500 (unit/s²), are used for the experiments. The experimental results of the four S-curves are shown in Table 1. From the data in the table, it can be seen that: after the introduction of the path length and speed constraints, it is verified that the four-segment and six-segment S-type speed planning curves can not reach the optimal speed within the constraints, and the five-segment speed planning can reach the optimal speed faster than seven-segment speed planning, and maintain the optimal feed rate for a longer time and a shorter time, which is a more ideal speed planning curve.