# Comparative Study on Effects of Input Configurations of Linear Quadratic Controller on Path Tracking Performance under Low Friction Condition

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{∞}control. Among these, MPC is the most widely adopted controller for PTC [11,13,14,17,19,20,21,22,23,24,25,26,27,28]. The advantage of these controllers is that it is easy to combine several types of control inputs. For example, the most commonly selected input configuration in LQR and MPC is the combination of front wheel steering (FWS), δ

_{f}, and longitudinal forces, ΔF

_{x}, [17,19,20,23,24,26]. The next is the combination of FWS and control yaw moment, ΔM

_{z}, [11,12,15,25,27]. If ΔM

_{z}is adopted in a particular input configuration, then a control allocation method or yaw moment distribution procedure is needed to convert it into longitudinal and lateral tire forces. In previous work, LQR and MPC with three input configurations, δ

_{f}, [δ

_{f}δ

_{r}]

^{T}, and [δ

_{f}ΔM

_{z}]

^{T}, were designed and compared under various speeds and friction conditions [11]. In another study, several path tracking controllers with FWS and 4WS were designed and compared with one another under high friction conditions [28,29]. Following the idea of these papers, the aim of this paper was to check the path tracking performance of several input configurations available in LQR.

_{x}or ΔM

_{z}should be included in the input configuration. After calculating these control inputs, these should be converted into longitudinal and later tire force by a control allocation method. This paper checks the effects of these actuators on path tracking performance.

_{f}and δ

_{r}, the control yaw moment, ΔM

_{z}, and the longitudinal control tire force, ΔF

_{x}. Among these, ΔF

_{x}can be converted into ΔM

_{z}by using force-moment equilibrium with geometric dimensions such as wheelbase and tread [17,19,20]. For this reason, IC#6 and IC#7 are neglected hereafter. As a result, three control inputs, δ

_{f}, δ

_{r}and ΔM

_{z}, constitute five input configurations from IC#1 to IC#5.

_{z}. However, 4WS or 4WIS cannot be used for IC#3 because they will break the optimality of FWS in IC#3. If active front steering (AFS) is assumed to be available, then 4WS can be used for IC#3. However, it is assumed that AFS is not available in this paper. For IC#4, only 4WIB and 4WID are available because 4WS is already set in IC#3. In fact, 4WIS cannot be used for the input configuration with FWS and 4WS, i.e., IC#1, IC#2, IC#3, and IC#4, because it will break the optimality of the control inputs of these input configurations. On the other hand, all the actuators, i.e., FWS, 4WS, 4WIS, 4WIB, and 4WID, are available for IC#5 because there are no actuators set to IC#5. The relationship between the input configurations and actuators is captured in a control allocation method.

_{z}for path tracking, a control allocation procedure was adopted. A simulation was conducted to verify the path tracking performance for each input configuration of LQR on a vehicle simulation package, CarSim. From the simulation results, it was shown that FWS or 4WS is enough for path tracking on low μ roads and that the control yaw moment or an additional actuator is not recommended as a control input of LQR.

## 2. Design of Path Tracking Controller with LQR

#### 2.1. Design of LQR

_{x}, is constant. The equations of motions of the model are derived as Equation (1) with the state variables [34]. The slip angles of front and rear wheels, α

_{f}and α

_{r}, are defined as Equation (2). In Equation (1), the lateral tire forces of front and rear wheels, F

_{yf}and F

_{yr}, are assumed to be linear to α

_{f}and α

_{r}, as shown in Equation (3), respectively. In Equation (1), ΔM

_{z}is the control yaw moment needed for VSC and PTC. By combining Equations (1)–(3), the linear equations for the 2-DOF bicycle model were obtained as Equation (4).

_{y}and e

_{φ}, are defined at the point P in Figure 2. In this paper, lookahead distance L

_{p}is calculated by Equation (5). Generally, k

_{v}is set between 1 and 2 s [41,44]. With L

_{p}in Figure 2, the point Q is obtained along the heading of a vehicle. From the point Q, the point R is obtained on the target path along the perpendicular direction to the heading of the vehicle. In this paper, the lateral offset and heading errors are calculated at the point R.

_{φp}and e

_{yp}, are derived as Equation (6) [46,47,48]. As given in (6), if the heading error is smaller than 10°, sine

_{φ}and φ

_{dp}can be approximated as e

_{φ}and φ

_{d}, respectively. In this paper, k

_{v}is set to a particular value smaller than 0.1. If v

_{x}is smaller than 20 m/s, then the lookahead point Q is located near the center of the front axle, which is nearly the same as the Stanley method [44].

**x**, and the disturbance,

**w**, are defined as Equation (9).

**u**can be set as given in Table 1. As shown in Table 2, there are no additional actuators for path tracking for IC#1 and IC#2, except for FWS and 4WS. On the other hand, in IC#3, IC#4, and IC#5, the control yaw moment ΔM

_{z}can be converted into the steering angle at the rear wheels or the braking and traction torques at each wheel by a relevant yaw moment distribution procedure, where braking and traction torques are generated by 4WIB and 4WID, respectively. The front steering angle, δ

_{f}, is available for all ICs. The rear steering angle, δ

_{r}, is available for all ICs, except for IC#1, while 4WIB and 4WID are available for IC#3, IC#4, and IC#5. As mentioned earlier, the objective of this paper was to compare these input configurations in terms of path tracking performance.

**B**

_{2i}, are given in Equation (11). In Equation (11),

**B**

_{2}(i) represents the i-th column of the input matrix

**B**

_{2}of Equation (10).

**Q**and

**R**

_{i}are obtained as Equation (14). In Equation (12), the weight ρ

_{i}is determined by Bryson’s rule, as given in Equation (15). In Equation (15), ξ

_{i}is the maximum allowable value of each term in Equation (12) [49]. For each input configuration, the control input

**u**

_{i}of LQR is obtained as Equation (16), where

**P**

_{i}is the solution of the Riccati equation.

#### 2.2. Control Allocation with the WLS-Based Method

_{z}is obtained in IC#3, IC#4, and IC#5, it should be converted into lateral and longitudinal tire forces generated by actuators. The actuators used to generate the lateral tire forces are FWS/RWS, 4WS, and 4WIS. The actuators used to generate the longitudinal tire forces are 4WIB and 4WID. The procedure needed to convert ΔM

_{z}into the tire forces of each wheel is called control allocation or yaw moment distribution. For control allocation in this paper, a WLS-based method was adopted [12,14,15,16,17,25].

_{z}, when ΔM

_{z}is positive [34]. The wheels in Figure 3 are numbered as 1, 2, 3, and 4 following the order of front left, front right, rear left, and rear right wheels. In Figure 3, ΔF

_{x}

_{1}to ΔF

_{x}

_{4}are the longitudinal tire forces generated by 4WIB and 4WID. If a longitudinal tire force is positive, then it is generated by 4WID. Otherwise, it is generated by 4WIB. Additionally, ΔF

_{y}

_{1}to ΔF

_{y}

_{4}stand for the lateral tire forces generated by FWS/RWS, 4WS, and 4WIS. Among them, ΔF

_{y}

_{1}and ΔF

_{y}

_{2}are generated by FWS, and ΔF

_{y}

_{3}and ΔF

_{y}

_{4}are generated by RWS or 4WS. To determine eight tire forces, the WLS-based method was used.

_{z}and tire forces is derived as Equation (17), where the elements of the vector

**g**are calculated as Equation (18). In Equation (17), there are no constraints on the lateral forces, ΔF

_{y}

_{1}, ΔF

_{y}

_{2}, ΔF

_{y}

_{3}, and ΔF

_{y}

_{4}. As a result, the corresponding steering angles, δ

_{1}, δ

_{2}, δ

_{3}, and δ

_{4}, can be freely generated by 4WIS. For this reason, Equation (17) cannot represent the relationship between the tire forces and the corresponding steering angles imposed by RWS and 4WS. For RWS and 4WS, the steering angles of the front or rear wheels should be identical to each other, which is represented by the constraints, in Equation (19).

_{F}, represents energy minimization, which is defined as Equation (20). In Equation (20), the radii of friction circles, μF

_{zi}, must be estimated. For the purpose in this paper, F

_{zi}was estimated using longitudinal and lateral accelerations [50]. In Equation (20),

**κ**is the vector of virtual weights, κ

_{i}. κ

_{i}serves to select the combination of each actuator [20,22]. The second part, J

_{C}, represents constraint satisfaction, which is defined as Equation (21). This is derived from Equation (17). In the previous study, Equation (17) should be satisfied in order to generate ΔM

_{z}. Under the condition, if ΔM

_{z}is large, then the lateral forces become much larger, which cannot be generated in real vehicles. To cope with this problem, the force-moment equilibrium Equation (17) is relaxed as Equation (21). By summing J

_{F}and J

_{C}with the tuning parameter η, the objective function J

_{CA}of WLS is obtained as Equation (22). In Equation (22), η is set to 1 or over it. Otherwise, the equality constraint, Equation (17), has no effect on J

_{CA}.

_{z}in

**W**takes the place of the friction circle constraint. The friction circle constraint can be used to measure the tire force margin (TFM), as given in Equation (24). If the TFM is small, the longitudinal and lateral tire forces approach its limits, which means an additional actuator used to generate these forces becomes less effective. As shown in Equation (24), TFM becomes much smaller under low μ conditions.

**q**and setting it to zero, Equation (25) is obtained. By solving Equation (25), the optimum solution of Equation (22) is algebraically obtained as (26). Generally, quadratic programming with an equality constraint, Equation (27), can be easily solved as Equation (28) by applying the Lagrange multiplier technique [12,14,15,16,17,25,35]. If the constraint, Equation (19), is added when applying FWS, RWS, and 4WS, then the quadratic programming with the objective function, Equation (22), and the equality constraint, Equation (19), is obtained. By expanding and rearranging Equation (22), Equation (29) is obtained, which is equivalent to the objective function of Equation (27). Therefore, the optimum solution of Equation (22) with the constraint Equation (19) is algebraically obtained as Equation (28).

_{z}can be selected by setting the virtual weights, κ

_{i}[34,52]. As shown in Equation (17), the vector

**q**has two parts, the lateral and longitudinal forces corresponding to the steering and braking/traction actuators. Thus, the virtual weights are set for the lateral and longitudinal forces. The vector of virtual weights corresponding to FWS, RWS, and 4WS/4WIS are given in Equations (30)–(32), respectively. In these equations, ε

_{i}is a very small value, i.e., 10

^{−4}, compared to 1, and ● represents the virtual weights corresponding to the longitudinal tire forces of 4WIB and 4WID. In Equation (17), the first two and next two elements in

**q**correspond to the front and rear wheels, respectively. Thus, Equation (30) represents the fact that the front wheel steering is available because ε

_{1}and ε

_{2}corresponding the front lateral forces are set to a very low value, i.e., 10

^{−4}. This makes the other lateral forces of

**q**

_{opt}, i.e., ΔF

_{y}

_{3}and ΔF

_{y}

_{4}, equal to zero. For the same reason, the virtual weights for RWS and 4WS/4WIS are set as Equations (31) and (32), respectively. It should be noted that the virtual weights of Equations (30)–(32) should be combined with the constraint, Equation (19), respectively. For example, if FWS is available, then Equations (19) and (30) are to be simultaneously used for optimization. Equation (19) guarantees that the front or rear steering angles are identical to each other, and Equation (30) guarantees that only the front lateral forces are generated by Equation (28).

_{z}. In these equations, ε

_{i}is a very small value, i.e., 10

^{−4}, compared to 1, and ∗ represents the virtual weights corresponding to the lateral tire forces of FWS, RWS, and 4WIS. As shown in Figure 3, only ΔF

_{x}

_{1}and ΔF

_{x}

_{3}should be generated if 4WIB is available and ΔM

_{z}is positive. This is represented by Equation (33). If 4WIB and 4WID are available for generating ΔM

_{z}, then no constraints imposed on the longitudinal forces are needed, regardless of the direction of ΔM

_{z}. This is represented by Equation (35). The virtual weights given in Equations (30)–(35) can be set according to actuators available for generating ΔM

_{z}, as given in Table 2. For example, if 4WS/4WIS, 4WIB, and 4WID are available, then Equations (32) and (35) should be combined. In this case, all the elements in the vector of virtual weights,

**κ**, have identical values. If 4WS is available, then Equation (32) should be imposed when solving Equation (26). As another example, Equations (31) and (34) should be combined if IC#3 in Table 2 is selected and RWS and 4WID are available.

**q**, as given in Equation (17) [31,34,52]. This is quite important when using rear-wheel steering in IC#2, IC#3, IC#4, and IC#5. When setting the weights ρ

_{i}in the LQ objective function of Equation (12), the steering angles of the front and rear wheels become identical to each other if the weights on the heading error and its rate, i.e., ρ

_{3}and ρ

_{4}in Equation (12), are set to a very small value, compared to those on the lateral offset error. As a consequence, β becomes large, which can make a vehicle lose its lateral stability [14]. Moreover, ride comfort also deteriorates. There are three ways to cope with this problem. The first is to set higher weights on the heading error and its rate in Equation (12). The second is to set a bound on ΔM

_{z}for IC#3, IC#4, and IC#5. The third is to set the virtual weights of the rear steering angles, i.e., ε

_{3}and ε

_{4}in Equations (31) and (32), higher. As a result, ΔF

_{x}

_{3}and ΔF

_{x}

_{4}become smaller than ΔF

_{x}

_{1}and ΔF

_{x}

_{2}in

**q**

_{opt}.

**q**

_{opt}in Equations (26) and (28), should be converted into a control input of each actuator. According to the sign of ΔF

_{xi}, the braking and traction torques, T

_{Bi}and T

_{Bi}, are calculated as Equations (36) and (37), respectively. In these equations, r

_{wi}, ω

_{i}, and ζ

_{i}are the tire radius, the rotational speed, and the ratio of reduction gear at the i-th wheel, respectively. The function h(●) represents the capacity curve of an electric motor.

_{yi}obtained in

**q**

_{opt}. In this paper, the steering angles were determined by using the definitions of the slip angle, Equation (2), and the linear lateral tire force, Equation (3). The linear lateral tire force, Equation (3), was rewritten as Equation (38). In Equation (38), σ is the parameter needed to tune the magnitude of the cornering stiffness, C

_{i}. In fact, σ is equivalent to a slip ratio [36]. For 4WS, the steering angles of the front and rear wheels are calculated as Equation (39) by combining Equation (2) with Equation (38) [11]. However, this does not hold for 4WIS because the slip angle, Equation (2), is defined not for 4WIS but for 4WS. For 4WIS, the slip angles of each wheel are calculated as Equation (40) by using the geometrical relationship and vehicle dimensions as given in Figure 3. By combining Equation (38) with Equation (40), the steering angles of 4WIS are calculated as Equation (41) [30,34]. In these formulations, β should be measured or estimated. In this paper, the Kalman Filter-based method was adopted to estimate β [53].

## 3. Performance Measures for Path Tracking Control

## 4. Simulation and Validation

_{x}and μ were set to 60 km/h and 0.4, respectively. In order to keep the vehicle speed constant, a built-in speed controller provided in CarSim was applied.

_{z}in order to make the rear steering angle small for IC#3, IC#4, and IC#5. For this purpose, ΔM

_{z}was limited to a certain value. This was performed after ΔM

_{z}was obtained from LQR. For IC#3 and IC#4, the maximum of ΔM

_{z}was limited to 2000 Nm [11]. The maximum lateral tire force from the F-segment sedan model in CarSim is 7500 N if μ is 1. This was 3000 N for this paper because μ was set to 0.4. From Table 2, the maximum available yaw moment was calculated as 3000 × 2 × (l

_{f}+ l

_{r}) = 18,000 Nm. Thus, the maximum of ΔM

_{z}was limited to 18,000 Nm for IC#5, in this paper. The maximum steering angles of the front and rear wheels were set to 30°.

**ξ**is the vector of the maximum allowable values, as given in Equation (15).

_{z}to the longitudinal and lateral tire forces. Table 6 and Table 7 show the parameters and gain elements of the controllers with IC#3 and IC#4, respectively, which were tuned such that ΔY was larger than −0.05 m. The maximum steering angles of the front and rear wheels were set to 30°. As mentioned earlier, the maximum ΔM

_{z}was limited to 2000 Nm.

_{z}, respectively. As shown in Table 5 and Table 8, there are also little differences between IC#1 and the first three rows of Table 8. This means that FWS itself was enough for path tracking on low μ roads without any additional actuators. As shown in Table 8 and Table 9, there are also little meaningful differences among measures for the LQRs with IC#3 and IC#4. This means that FWS or 4WS itself has a large effect on path tracking performance. As a result, there is a small tire force margin to improve the performance, as shown in Figure 6b. Moreover, those actuator combinations used a relatively small ΔM

_{z}within the given limit, as shown in the last column of Table 8 and Table 9, Max.|ΔM

_{z}|. Even though ΔM

_{z}was saturated to 2000Nm, the controllers showed good path tracking performance. These results confirm that additional actuators to FWS or 4WS are not effective due to the small tire force margin.

_{z}to the longitudinal and lateral tire forces. Moreover, there were no FWS or 4WS in IC#5. Table 10 shows the parameters and gain elements of the controllers, which were tuned such that ΔY was larger than −0.05 m. The maximum steering angles of the front and rear wheels were set to 30°. As mentioned earlier, the maximum ΔM

_{z}was limited to 18,000 Nm. In this case, the virtual weights on the rear steering wheels were set higher because it was necessary to limit the rear steering angles in order to keep β as small as possible. The vectors of virtual weights for each actuator combination are given in Table 10.

_{z}. As shown in Table 11, there are little meaningful differences between actuator combinations. A notable feature of IC#5 is that all the measures were slightly improved by several actuator combinations used for control allocation. ΔX was clearly improved for all actuator combinations. For example, using only FWS to generate ΔM

_{z}for IC#5 showed better ΔX than IC#1 by comparing the first rows of Table 5 and Table 11. However, every actuator combination except FWS and 4WS in IC#5 needed more than two actuators to generate ΔM

_{z}. Moreover, they require a tedious and time-consuming tuning process on several parameters and weights of the controllers, compared to IC#1 and IC#2. To sum up the above results, it can be concluded that the LQR with IC#1 or IC#2 was quite effective enough, and no additional actuators are not needed for path tracking on low μ roads.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

4WS | 4-wheel steering |

4WIS | 4-wheel independent steering |

4WIB | 4-wheel independent braking |

4WID | 4-wheel independent drive |

AFS | active front steering |

FWS | front wheel steering |

MASSA | maximum absolute side-slip angle |

RWS | rear wheel steering |

RWIS | rear wheel independent steering |

TFM | tire force margin |

WLS | weighted least square |

C_{f}, C_{r} | cornering stiffness of front and rear tires (N/rad) |

C_{i} | cornering stiffness of i-th wheel (N/rad) |

e_{y}, e_{φ} | lateral offset error (m) and heading error (rad) |

e_{yp}, e_{φp} | lateral offset error (m) and heading error (rad) obtained from lookahead |

F_{xi}, F_{yi}, F_{zi} | longitudinal, lateral, and vertical tire forces of i-th wheel (N) |

F_{yf}, F_{yr} | front and rear lateral tire forces in the 2-DOF bicycle model (N) |

g | vector used for the equality constraint in WLS-based method |

h() | capacity curve of an electric motor |

H | matrix used for the constraint on RWS and 4WS in the WLS-based method |

I_{z} | yaw moment of inertial (kg·m^{2}) |

J_{i} | LQ objective function for the input configuration IC#i |

K_{LQR,i} | gain matrix of LQR for input configuration IC#i |

k_{v} | velocity gain for lookahead distance |

L_{p} | lookahead distance (m) |

l_{f}, l_{r} | distance from CoG to front and rear axles (m) |

m | vehicle total mass (kg) |

OS% | percentage of overshoot in the lower lane of the target path |

q | vector of tire forces as a solution to the WLS-based method |

r_{wi} | radius of i-th wheel (m) |

T_{Bi}, T_{Di} | braking and traction torques applied at i-th wheel (N·m) |

t_{f}, t_{r} | half of track widths of front and rear axles (m) |

v_{x}, v_{y} | longitudinal and lateral velocities of CoG of a vehicle (m/s) |

W | weighting matrix of the WLS-based method |

X(∗), Y(∗) | x- and y-positions of the point ∗ on the target path and vehicle trajectory |

Y_{ref}(X) | y-position of the target path with respect to X |

y | lateral offset of a vehicle |

y_{d}, y_{dp} | desired lateral offset obtained without and with lookahead |

α_{f}, α_{r} | tire slip angles of front and rear wheels (rad) |

α_{i} | tire slip angle of i-th wheel (rad) |

β | side-slip angle of CoG of a vehicle (rad) = tan^{−1}(v_{y}/v_{x}) ≈ (v_{y}/v_{x}) |

δ_{f}, δ_{r} | front and rear steering angles (rad) |

δ_{i} | steering angle of i-th wheel (rad) |

ε_{i} | virtual weights on corresponding lateral and longitudinal tire forces |

ΔF_{xi}, ΔF_{yi} | control longitudinal and lateral forces generated by an actuator (N) |

ΔF_{x} | longitudinal force as a control input in LQR (N) |

ΔM_{z} | control yaw moment as a control input in LQR (N·m) |

ΔX, ΔY | differences between x- and y-positions at the peak points of the target path |

ΔDX, ΔSX | response and settling delays of vehicle trajectory with respect to target path |

γ, γ_{d} | real and reference yaw rates (rad/s) |

η | tuning parameter on relaxation term of equality constraint |

χ | curvature at a particular point on a target path |

κ | virtual weight on the longitudinal and lateral tire forces |

κ | vector of virtual weights |

ξ_{i} | the maximum allowable value of i-th term in LQ objective function |

ξ | vector of the maximum allowable values |

ϕ_{i} | equivalent slip angle of i-th wheel calculated from control lateral tire force |

φ | heading angle of a vehicle |

φ_{d}, φ_{dp} | desired heading angle obtained without and with lookahead |

ψ_{ref}(χ) | heading angle of the target path with respect to X |

μ | tire-road friction coefficient |

ω_{i} | rotational speed of i-th wheel (rad/s) |

ρ_{i} | weight on i-th term in LQ objective function |

σ | equivalent slip ratio for slip angle calculation |

ζ_{i} | ratio of reduction gear of i-th wheel |

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**Figure 5.**Simulation results for IC#1 and IC#2: (

**a**) steering angles, (

**b**) trajectories, (

**c**) side–slip angles.

**Figure 6.**Simulation results of tire forces for IC#1 and IC#2: (

**a**) slip angles vs. lateral tire forces, (

**b**) tire force margin.

Controller | Input Configuration | Actuators | μ | Reference |
---|---|---|---|---|

PID, SMC, LQR, MPC | [δ_{f}], [δ_{f} δ_{r}] | FWS, 4WS | 0.4 | [28] |

MPC, LQR, FF | [δ_{f}], [δ_{f} δ_{r}], [δ_{f} ΔM_{z}] | FWS, 4WS | 0.3 | [11] |

MPC, SMC | [δ_{f} ΔM_{z}], [δ_{f} ΔF_{x}] | FWS, 4WIB, 4WID | 0.4 | [20] |

LQR | [δ_{f} ΔM_{z}] | FWS, 4WID | 0.4 | [12] |

LMI-based SMC | FWS, 4WIB, 4WID | 0.3 | [15] | |

MPC | FWS, 4WIB, 4WID | 0.3 | [25] | |

MPC | FWS, 4WID | 0.5 | [27] | |

MPC | [δ_{f} δ_{r} ΔM_{z}] | 4WIS | 0.25, 0.5 | [13] |

MPC | 4WIS, 4WID | 0.25, 0.6 | [14] | |

LPV/H(control) | 4WIS, 4WID | 0.3 | [16] | |

MPC | [δ_{f} ΔF_{x}] | FWS, 4WID | 0.4 | [17] |

FWS, 4WIB, 4WID | 0.6 | [19] | ||

FWS | 0.2 | [23] | ||

FWS, 4WIB | 0.3 | [24,26] | ||

LQR | [δ_{f} δ_{r}] | 4WIS | 0.4, 0.65 | [18] |

LQR, FF | 4WS | 0.25, 0.5 | [21] | |

MPC | [δ_{f} ΔM_{z} ΔF_{x}] | FWS, 4WIB, 4WID | 0.3 | [22] |

Input Configurations | Control Inputs | Available Actuators |
---|---|---|

IC#1 | u_{1} = δ_{f} | FWS |

IC#2 | u_{2} = [δ_{f} δ_{r}]^{T} | 4WS |

IC#3 | u_{3} = [δ_{f} ΔM_{z}]^{T} | FWS, RWS, RWIS, 4WID, 4WIB |

IC#4 | u_{4} = [δ_{f} δ_{r} ΔM_{z}]^{T} | 4WS, 4WID, 4WIB |

IC#5 | u_{5} = ΔM_{z} | FWS, 4WS, 4WIS, 4WID, 4WIB |

IC#6 | u_{6} = [δ_{f} ΔF_{x}]^{T} | FWS, 4WID, 4WIB |

IC#7 | u_{7} = [δ_{f} ΔF_{x} ΔM_{z}]^{T} | FWS, 4WID, 4WIB |

Parameter | Value | Parameter | Value |
---|---|---|---|

m_{s} | 1823 kg | I_{z} | 6286 kg·m^{2} |

C_{f} | 42,000 N/rad | C_{r} | 62,000 N/rad |

l_{f} | 1.27 m | l_{r} | 1.90 m |

t_{f} | 0.80 m | t_{r} | 0.80 m |

k_{v} | Gains | |
---|---|---|

IC#1 | 0.1 | ξ = [0.54, 5.00, 0.30, 10.00, 0.05] |

IC#2 | 0.1 | ξ = [0.52, 2.00, 0.20, 0.70, 0.05, 0.02] |

ΔX (m) | ΔY (m) | OS% | ΔDX (m) | ΔSX (m) | MASSA (deg) | |
---|---|---|---|---|---|---|

IC#1 | 2.09 | −0.025 | 0.87 | 8.77 | 4.34 | 0.59 |

IC#2 | 1.99 | −0.026 | 0.66 | 8.35 | 3.88 | 0.92 |

k_{v} | η | Gains | |
---|---|---|---|

+4WID | 0.1 | 10 | ξ = [0.530, 2.000, 0.200, 1.000, 0.050, 1000.0] |

+4WIB | 0.1 | 10 | ξ = [0.540, 2.500, 0.130, 1.000, 0.050, 1500.0] |

+4WID + 4WIB | 0.1 | 10 | ξ = [0.500, 1.500, 0.100, 0.500, 0.050, 2000.0] |

RWS | 0.1 | 10 | ξ = [0.540, 2.000, 0.300, 1.000, 0.050, 500.0] |

+4WID | 0.1 | 10 | ξ = [0.530, 2.000, 0.300, 1.000, 0.050, 1000.0] |

+4WIB | 0.1 | 10 | ξ = [0.510, 2.500, 0.150, 1.000, 0.050, 1500.0] |

+4WID + 4WIB | 0.1 | 10 | ξ = [0.500, 2.000, 0.100, 1.000, 0.050, 2000.0] |

RWIS | 0.1 | 10 | ξ = [0.530, 3.000, 0.300, 1.000, 0.050, 500.0] |

+4WID | 0.1 | 10 | ξ = [0.520, 2.000, 0.200, 1.000, 0.050, 800.0] |

+4WIB | 0.1 | 10 | ξ = [0.580, 2.700, 0.150, 0.300, 0.050, 800.0] |

+4WID + 4WIB | 0.1 | 10 | ξ = [0.520, 2.500, 0.200, 1.000, 0.050, 800.0] |

k_{v} | η | Gains | |
---|---|---|---|

+4WID | 0.1 | 10 | ξ = [0.530, 3.000, 0.250, 0.200, 0.050, 0.020, 500.0] |

+4WIB | 0.1 | 10 | ξ = [0.530, 3.500, 0.300, 0.300, 0.050, 0.010, 500.0] |

+4WID + 4WIB | 0.1 | 10 | ξ = [0.520, 3.000, 0.300, 0.200, 0.050, 0.020, 500.0] |

ΔX (m) | ΔY (m) | OS% | ΔDX (m) | ΔSX (m) | MASSA (deg) | Max|ΔM_{z}|(Nm) | |
---|---|---|---|---|---|---|---|

+4WID | 2.01 | −0.031 | 0.81 | 8.58 | 4.25 | 0.59 | 2000 |

+4WIB | 1.97 | −0.029 | 0.24 | 8.15 | 4.63 | 0.57 | 2000 |

+4WID + 4WIB | 1.98 | −0.036 | 0.75 | 8.74 | 4.61 | 0.59 | 2000 |

RWS | 1.79 | −0.035 | 0.32 | 8.63 | 5.28 | 0.83 | 595 |

+4WID | 2.11 | −0.031 | 0.76 | 8.81 | 4.97 | 0.86 | 2000 |

+4WIB | 2.11 | −0.024 | 0.85 | 8.72 | 4.30 | 0.69 | 2000 |

+4WID + 4WIB | 2.04 | −0.027 | 0.67 | 8.84 | 4.73 | 0.69 | 2000 |

RWIS | 2.08 | −0.031 | 0.66 | 8.70 | 4.69 | 0.74 | 574 |

+4WID | 1.95 | −0.036 | 0.47 | 8.36 | 4.43 | 0.61 | 1428 |

+4WIB | 2.14 | −0.026 | 0.79 | 8.68 | 5.87 | 3.43 | 1365 |

+4WID + 4WIB | 2.06 | −0.031 | 0.60 | 8.64 | 4.57 | 0.64 | 1446 |

IC#4 | ΔX (m) | ΔY (m) | OS% | ΔDX (m) | ΔSX (m) | MASSA (deg) | Max|ΔM_{z}|(Nm) |
---|---|---|---|---|---|---|---|

+4WID | 2.09 | −0.021 | 0.62 | 8.42 | 4.35 | 1.57 | 517 |

+4WIB | 2.01 | −0.031 | 0.60 | 8.48 | 4.33 | 0.44 | 560 |

+4WID + 4WIB | 2.06 | −0.023 | 0.69 | 8.40 | 4.01 | 1.41 | 520 |

k_{v} | η | Gains | |
---|---|---|---|

FWS | 0.06 | 1 | ξ = [0.820, 0.800, 0.200, 0.300, 1000.0] |

+4WID | 0.06 | 1 | ξ = [0.730, 0.600, 0.200, 0.100, 1000.0] |

+4WIB | 0.06 | 1 | ξ = [0.700, 0.600, 0.200, 0.300, 1000.0] |

+4WID + 4WIB | 0.06 | 1 | ξ = [0.460, 0.400, 0.300, 0.200, 1000.0] |

4WS | 0.06 | 10 | ξ = [0.100, 0.050, 0.020, 0.020, 1500.0]κ = [10^{−4}, 10^{−4}, 5 × 10^{−4}, 5 × 10^{−4}, ●, ●, ●, ●] |

+4WID | 0.06 | 10 | ξ = [0.110, 0.050, 0.050, 0.020, 1500.0]κ = [10^{−4}, 10^{−4}, 5 × 10^{−4}, 5 × 10^{−4}, ●, ●, ●, ●] |

+4WIB | 0.06 | 10 | ξ = [0.085, 0.050, 0.010, 0.010, 1500.0]κ = [10^{−4}, 10^{−4}, 5 × 10^{−4}, 5 × 10^{−4}, ●, ●, ●, ●] |

+4WID + 4WIB | 0.06 | 10 | ξ = [0.085, 0.050, 0.010, 0.010, 1500.0]κ = [10^{−4}, 10^{−4}, 5 × 10^{−4}, 5 × 10^{−4}, ●, ●, ●, ●] |

4WIS | 0.06 | 1 | ξ = [0.300, 0.300, 0.060, 0.050, 800.0]κ = [10^{−4}, 10^{−4}, 3 × 10^{−3}, 3 × 10^{−3}, ●, ●, ●, ●] |

+4WID | 0.06 | 1 | ξ = [0.240, 0.240, 0.020, 0.010, 1000.0]κ = [10^{−4}, 10^{−4}, 3 × 10^{−3}, 3 × 10^{−3}, ●, ●, ●, ●] |

+4WIB | 0.06 | 1 | ξ = [0.200, 0.200, 0.020, 0.030, 600.0]κ = [10^{−4}, 10^{−4}, 3 × 10^{−3}, 3 × 10^{−3}, ●, ●, ●, ●] |

+4WID + 4WIB | 0.06 | 1 | ξ = [0.160, 0.150, 0.200, 0.016, 1000.0]κ = [10^{−4}, 10^{−4}, 3 × 10^{−3}, 3 × 10^{−3}, ●, ●, ●, ●] |

ΔX (m) | ΔY (m) | OS% | ΔDX (m) | ΔSX (m) | MASSA (deg) | Max|ΔM_{z}|(Nm) | |
---|---|---|---|---|---|---|---|

FWS | 1.15 | −0.016 | 0.59 | 8.48 | 4.46 | 0.59 | 10,524 |

+4WID | 1.17 | −0.025 | 0.42 | 8.25 | 4.49 | 0.58 | 12,230 |

+4WIB | 1.12 | −0.005 | 0.19 | 7.69 | 3.92 | 0.57 | 11,448 |

+4WID + 4WIB | 1.18 | −0.039 | 0.80 | 8.91 | 5.06 | 0.63 | 17,024 |

4WS | 1.22 | 0.061 | 0.84 | 8.37 | 3.90 | 0.67 | 18,000 |

+4WID | 1.27 | 0.040 | 0.85 | 8.49 | 4.39 | 0.66 | 18,000 |

+4WIB | 0.85 | 0.063 | 0.06 | 7.67 | 3.78 | 0.64 | 18,000 |

+4WID + 4WIB | 1.22 | 0.042 | 0.80 | 8.65 | 4.81 | 0.66 | 18,000 |

4WIS | 1.97 | 0.020 | 0.89 | 9.20 | 4.95 | 0.76 | 18,000 |

+4WID | 1.00 | −0.005 | -0.01 | 8.16 | 4.87 | 0.75 | 18,000 |

+4WIB | 1.45 | −0.003 | 0.18 | 8.08 | 5.50 | 1.53 | 18,000 |

+4WID + 4WIB | 1.42 | −0.009 | 0.82 | 9.23 | 5.40 | 0.78 | 18,000 |

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**MDPI and ACS Style**

Park, M.; Yim, S.
Comparative Study on Effects of Input Configurations of Linear Quadratic Controller on Path Tracking Performance under Low Friction Condition. *Actuators* **2023**, *12*, 153.
https://doi.org/10.3390/act12040153

**AMA Style**

Park M, Yim S.
Comparative Study on Effects of Input Configurations of Linear Quadratic Controller on Path Tracking Performance under Low Friction Condition. *Actuators*. 2023; 12(4):153.
https://doi.org/10.3390/act12040153

**Chicago/Turabian Style**

Park, Manbok, and Seongjin Yim.
2023. "Comparative Study on Effects of Input Configurations of Linear Quadratic Controller on Path Tracking Performance under Low Friction Condition" *Actuators* 12, no. 4: 153.
https://doi.org/10.3390/act12040153