Model predictive control–based steering control algorithm for steering efficiency of a human driver in all-terrain cranes

Abstract

All-terrain cranes with multi-axles have large inertia and long distances between the axles that lead to a slower dynamic response than normal vehicles. This has a significant effect on the dynamic behavior and steering performance of the crane. Therefore, the purpose of this study is to develop an optimal steering control algorithm with a reduced driver steering effort for an all-terrain crane and to evaluate the performance of the algorithm. For this, a model predictive control technique was applied to an all-terrain crane, and a steering control algorithm for the crane was proposed that could reduce the driver’s steering effort. The steering performances of the existing steering system and the steering system applied with the newly developed algorithm were compared using MATLAB/Simulink and ADAMS with a human driver model for reasonable performance evaluation. The simulation was performed with both a double lane change scenario and a curved-path scenario that are expected to happen in road-steering mode.

Introduction

All-terrain cranes, the most typical example of mobile construction equipment, have an excellent lifting capacity and the capability to operate both off-road and on-road. However, its large inertia and long axle-distance slow down the dynamic response during steering, and its high center of mass can cause rollover. Hence, a technology to improve the steering performance needs to be developed. Various studies regarding the steering control of mobile construction equipment, including the all-terrain crane, are being actively carried out.

Wang et al. ¹ proposed a proportional control strategy by using zero side-slip angles at the mass center and by evaluating the steering performance of a heavy vehicle with three axles in different steering modes. A multi-mode steering control scheme using a cross-coupled control was developed to improve the steering accuracy of a large-scale transportation vehicle. ² A self-tuning proportional–integral–derivative (PID) control with fuzzy logic was presented for an electro-hydraulic steering system of a multi-axle crane. ³ In the research of Noh et al., ⁴ turning stability was studied for a multi-axle steering vehicle by means of the Ackerman theory in the case of a high-speed running mode. Qin et al. ⁵ developed a multi-bridge steering system for an all-terrain crane. In their study, a hydraulic model and a multi-body model for a multi-bridge steering system were designed using AMESim and ADAMS software, respectively. Gao and Li ⁶ proposed a novel steering algorithm for a multi-axle vehicle by combining the traditional Ackerman steering and the Skid steering method, and explored the turning characteristics. Zhang et al. ⁷ developed a novel fuzzy observer-based steering control algorithm for path tracking in autonomous vehicles. The effectiveness of the proposed fuzzy observer-based output feedback controller was evaluated in Carsim/Matlab joint simulation environment. Cao et al. ⁸ proposed a new vehicle path-following strategy of the steering driver model using general predictive control method. The purpose of the proposed active steering control algorithm by the authors is to assist the driver to follow the desired path efficiently. The algorithm was evaluated in some typical conditions such as strong crosswind in standard double lane change.

However, the above studies were based on the existing PID control, Ackerman theory, and observer-based control, and the purpose of the studies generally focused on improving the desired path tracking performance of the vehicles. In addition, the previous studies did not consider the driver’s steering effort to affect the driver’s fatigue while driving.

Therefore, this study proposes a steering control strategy that considers the driver’s characteristics in order to improve the steering performance (reducing steering effort of driver) of a 120-ton (maximum lifting capacity) all-terrain crane with multiple axles. For this, a simplified crane model for the crane steering system was derived in terms of lateral and yaw dynamics, and its error dynamics were modeled. Moreover, the optimal steering control algorithm that would reduce the driver’s steering effort was developed using a model predictive control (MPC) technique. To evaluate the developed control algorithms, a virtual steering input of human driver was computed using a predictive human driver model with the MPC technique with the assumption that the road information can be obtained from a sensor such as camera for obtaining errors for feedback control. Then, a co-simulation linked with both MATLAB/Simulink and ADAMS was performed under the scenarios of a double lane change and a curved path in road-steering mode. Based on the analysis results, the steering system applied with the proposed algorithm and the existing steering system were compared from the perspective of steering effort and dynamic stability.

The article is organized as follows. “Conventional steering principle for all-terrain crane” section describes a conventional steering principle for an all-terrain crane. “Modeling of an all-terrain crane steering system” section describes the modeling of the steering system. “Design of a human driver model” section describes the design of the human driver model. “Design of control algorithm” section describes the design of the steering control algorithms. “Evaluation of steering performance” section describes the evaluation of the steering performance. Finally, concluding remarks are provided in the “Conclusion” section.

Conventional steering principle for all-terrain crane

The conventional steering principle for existing all-terrain cranes is as follows. As the motion of the second of a crane is mechanically constrained by the first axle, the steering angle at the second axle is subject to that of the first axle (i.e. the driver’s steering input angle). The relationship between these steering angles is provided in Figure 1 and equation (1). The steering angles at the third, fourth, and fifth axles are independently determined according to the steering mode and driving speed of the crane.

Figure 1.

Correlation between the steering angles of the first and second axle.

Typical steering modes include the all-wheel steering mode, the road-steering mode, the crab steering mode, the reduced swing-out mode, and the independent steering mode. Among these, the road-steering mode is used for general driving on public roads

$${\delta }_2=r_{\mathrm{str}}{\delta }_1$$

(1)

where ${\delta }_1$ and ${\delta }_2$ represent the first and second steering angles, respectively, and $r_{\mathrm{str}}$ represents a constant ratio to relate ${\delta }_1$ and ${\delta }_2$ (about 0.5). The conventional steering principle of the existing system uses the Ackerman theory in the road-steering mode, arranging the steering pole on an extension line of the third axle. The driving speed of the crane affects the steering strategy in the road-steering mode. As shown in Figure 2, the third axle is fixed at a driving speed of 0–30 km/h, the third and fourth at 30–60 km/h, and the third, fourth, and fifth axles at a speed higher than 60 km/h.

Figure 2.

Ackerman theory–based steering strategy of the third, fourth, and fifth axles at various speeds in road-steering mode.

As it can be seen in the Figure 2, the third axle was designed to be fixed for all velocity conditions in the conventional steering principle of actual cranes. This is to ensure driving safety by creating a large turning radius from the axles that are fixed by speed, because a larger lateral acceleration occurs even at a small yaw rate as the speed increases. Figure 3 shows the correlation between the steering angle at the first axle and those at the third, fourth, and fifth axles at various driving speeds in road-steering mode.

Figure 3.

Relationship between steering angle at the first axle and those at the third, fourth, and fifth axles in road-steering mode: top: 0–30 km/h, middle: 30–60 km/h, bottom: >60 km/h.

This article proposes a steering strategy that improves the steering performance of an all-terrain crane and reduces the driver’s steering effort in road-steering mode.

Modeling of an all-terrain crane steering system

Simplified model for a multi-axle crane steering system

In this study, a simplified steering system model for an all-terrain crane with five axles (Figure 4) was derived to analyze its lateral motion using a bicycle model that has been widely used for an analysis of vehicle dynamics (lateral, yaw, side slip, etc.), controller design, and control performance evaluation. Several assumptions applied for the mathematical model derivation are listed as follows:

Figure 4.

Simplified modeling of a crane using a bicycle model.

Assumptions for mathematical modeling of a crane steering system are as follows:

1. Longitudinal speed is constant.

2. Body slip angle is sufficiently small.

3. Left slip angles are identical with right slip angles.

As the figure illustrates, both wheels at each axle of the crane are represented by a single wheel on the center line of the axle.

In the figure, $L_i(i=1~5)$ refers to the distance from the mass center to each axle, $F_{y,i}(i=1~5)$ is the lateral force, ${\delta }_{L,i}$ is the steering angle on the left wheel, and ${\delta }_{R,i}$ is the steering angle on the right wheel. The term ${\delta }_{c,i}$ represents the equivalent steering angle on the center line (i.e. equivalent center steering angle) and can be calculated as below

$${\delta }_{c,i}=\frac{{\delta }_{L,i}+{\delta }_{R,i}}{2},(i=1,2,3,4,5)$$

(2)

Based on the formula above, the state and the state-space model showing the lateral and yaw dynamics of the crane are given in equations (3) and (4)–(6), respectively ⁹

$$x={\left[\begin{array}{cccc}y\hfill & \stackrel{·}{y}\hfill & \psi \hfill & \stackrel{·}{\psi }\hfill \end{array}\right]}^T$$

(3)

where y and $\stackrel{·}{y}$ are the lateral displacement and velocity of the crane, respectively. The terms $\psi$ and $\stackrel{·}{\psi }$ are the yaw angle and yaw rate of the crane, respectively

$$\mathrm{mv}(\stackrel{·}{\psi }+\stackrel{·}{\beta })=F_{y,1}+F_{y,2}+F_{y,3}+F_{y,4}+F_{y,5}$$

(4)

where $m$ , $v$ , $\beta$ , and $F_y$ represent the crane mass, longitudinal velocity, sideslip angle, and lateral tire force, respectively

$$I\stackrel{··}{\psi }=L_1{F}_{y,1}+L_2{F}_{y,2}-L_3{F}_{y,3}-L_4{F}_{y,4}-L_5{F}_{y,5}$$

(5)

where $I$ is the moment of inertia of the crane. Based on the aforementioned dynamic equations and the assumption that the crane’s lateral tire force has a linear relationship with the slip, the linear state-space equation can be derived as follows

$$\stackrel{·}{x}=\mathrm{Ax}+B{\delta }_c$$

(6)

where $A=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& a_{22}\hfill & 0\hfill & a_{24}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& a_{42}\hfill & 0\hfill & a_{44}\hfill \end{array}\right]$ , $a_{22},a_{24},a_{42},\mathrm{and}{a}_{44}$ are defined in Appendix 2. The matrix B is defined as $B=\left[\begin{array}{ccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{2C_1/m}& 2C_2/m\hfill & 2C_3/m\hfill & 2C_4/m\hfill & 2C_5/m\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{2L_1{C}_1/I}& 2L_2{C}_2/I\hfill & 2L_3{C}_3/I\hfill & 2L_4{C}_4/I\hfill & 2L_5{C}_5/I\hfill \end{array}\right]$ .

$C_i(i=1~5)$ is the cornering stiffness of the tire at the ith axle and ${\delta }_c=[\begin{array}{ccccc}{\delta }_{c,1}\hfill & {\delta }_{c,2}\hfill & {\delta }_{c,3}\hfill & {\delta }_{c,4}\hfill & {\delta }_{c,5}\hfill \end{array}{]}^T$ . The dynamic model of the crane derived above was implemented using Matlab/Simulink and is used to analyze the dynamic behavior of the crane, as well as the steering performance of the proposed steering control system.

Error dynamics for steering control

Information about the state error (i.e. the lateral and yaw angle errors), as shown in Figure 5 and equations (7) and (8), is necessary for the design of a steering control algorithm.

Figure 5.

Lateral and yaw angle errors for error dynamics.

The X–Y and x–y coordinates described in Figure 5 are the crane’s global and local coordinates, respectively. The local coordinate is based on the mass center of the crane. The term e₁ represents the lateral displacement error (i.e. the difference between the lateral position $y$ and its desired value $y_{\mathrm{des}}$ ) and e₂ is the yaw angle error (i.e. the difference between the yaw angle $\psi$ and its desired value ${\psi }_{\mathrm{des}}$ ), all of which are defined with the mass center as their basis

$$e_1=y-y_{\mathrm{des}}$$

(7)

$$e_2=\psi -{\psi }_{\mathrm{des}}$$

(8)

The state vector e for error dynamics with e₁ and e₂ is as follows

$$e={\left[\begin{array}{cccc}{e}_1\hfill & {\stackrel{·}{e}}_1\hfill & e_2\hfill & {\stackrel{·}{e}}_2\hfill \end{array}\right]}^T$$

(9)

where ${\stackrel{·}{e}}_1$ and ${\stackrel{·}{e}}_2$ are the error rates for $e_1$ and $e_2$ , respectively. For reference, the desired yaw rate ( ${\stackrel{·}{\psi }}_{\mathrm{des}}$ ) defined using the radius of curvature (R_r) is given below

$${\stackrel{·}{\psi }}_{\mathrm{des}}=\frac{v}{R_r}$$

(10)

where R is the radius of curvature. As the rate of curvature change in public driving roads is not sufficiently large, the desired yaw rate ${\stackrel{·}{\psi }}_{\mathrm{des}}$ and its acceleration ${\stackrel{··}{\psi }}_{\mathrm{des}}$ were assumed to be zero because these values are close to zero. The error dynamics of state-space equation form were derived as shown below ¹⁰

$$\stackrel{·}{e}=A_{e}e+B_e{\delta }_c$$

(11)

where $A_e=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& a_{e22}\hfill & a_{e23}\hfill & a_{e24}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& a_{e42}\hfill & a_{e43}\hfill & a_{e44}\hfill \end{array}\right]$ and elements of the matrix are defined in Appendix 2. The matrix B_e is same as B defined in equation (6). The error states defined in this study can be computed based on the reference path. The reference path described in Figure 5 can be derived using a environment sensor such as camera. Based on the derived reference path, the desired relative values such as lateral displacement and yaw angle can be computed considering vehicle states.

Design of a human driver model

In order to conduct reasonable performance evaluation of the steering control algorithm virtually, the human driver model should be used. Therefore, the predictive driver model was used for performance evaluations in this study and the steering input from the human driver model was considered as an actual human driver for evaluation of the driver’s effort. In general, a driver displays three steering characteristics while driving. First, the steering input is applied after perceiving the vehicle’s preview information and state. Second, the steering input applied here is a prediction-based input. Third, the update frequency of steering input with the changing environment varies with the proficiency and sight of the driver. The driver’s steering characteristic about time delay (time delay characteristic is realized by update frequency) is one of the most important characteristic caused by brain response and neuromuscular system. In this study, the update frequency is defined as 0.1 s. In this study, the human driver model has been used to conduct reasonable performance evaluation of the steering control algorithm developed. Figure 6 shows the driver’s steering input behavioral characteristics.

Figure 6.

Driver’s steering behavior.

In the figure, a preview distance $(L_p)$ represents the distance from driver’s position to a preview point. The preview point can be referred as the maximum viewpoint of the driver to identify a desired path in advance. Because the preview distance is varied with the vehicle velocity, it can be calculated using the vehicle speed $v$ and preview time $t_p$ as shown in equation (12)

$$L_p=t_{p}v$$

(12)

In the driver model, the lateral error $(e_{1,d})$ and yaw angle error $(e_{2,d})$ at the preview point were defined as shown in Figure 7 to reflect the driver’s steering behavioral characteristics.^11–14

Figure 7.

Lateral and yaw angle errors for the driver model.

Also, the lateral error and yaw angle error were computed using the tangent line on a reference path near the preview point. The dynamics of these errors can be defined as below

$$e_{d\_p}={\left[\begin{array}{cc}{e}_{1,d}\hfill & e_{2,d}\hfill \end{array}\right]}^T$$

(13)

$${\stackrel{·}{e}}_{d\_p}=A_{e,d}{e}_{d\_p}+B_{e,d}{\delta }_{c,1}+G_{e,d}{\stackrel{·}{\psi }}_{\mathrm{des}}$$

(14)

$$e_{d\_f}={[e_{1,d},{\stackrel{·}{e}}_{1,d},e_{2,d},{\stackrel{·}{e}}_{2,d}]}^T$$

(15)

where $A_{e,d}=\left[\begin{array}{cc}0\hfill & v\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ , $B_{e,d}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{v/vLL}\end{array}\right]$ , and $G_{e,d}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-1}\end{array}\right]$ . The term $v$ is the crane’s longitudinal velocity and $L$ is the distance between the first and third axles (i.e. $L=L_1+L_3$ ). The matrices $A_{e,d}$ , $B_{e,d}$ , and $G_{e,d}$ have been discretized for the MPC algorithm. The detailed derivation process of the error dynamics can be found in Mayne et al. ¹⁵

The error dynamics derived above, $e_{1,d}$ and $e_{2,d}$ defined in Figure 7, are used as the input values for the MPC-based driver model and the MPC-based steering model (see Figure 8).

Figure 8.

Structure of the MPC-based steering control system.

The error states in equation (13) have been used for the human driver model and steering model because the purpose of the steering control algorithm is to reduce the driver’s steering effort for steering efficiency (the error states used for the human driver model also have been used for the steering model). The steering model described in Figure 8 needs more states such as derivatives of $e_{1,d}$ and $e_{2,d}$ than the human driver model. The predictive driver’s steering input can be computed by the human driver model. Three representative driving characteristics (using preview information, predictive inputs, and input date frequency) have been quantized using the parameters such as preview time and update frequency.

Design of control algorithm

Structure of the control system

The steering control algorithm proposed in this study uses a model predictive human driver model to derive the realistic human driver’s steering angle of the first axle. Specifically, the steering angles at the third, fourth, and fifth axles were computed according to the human driver’s input using MPC algorithm (however, the steering angle at the second axle was determined by the driver’s input based on equation (2)). In addition, the error dynamics and the MPC algorithm are used to derive the optimal steering angles of the third, fourth, and fifth axles, which allow individual steering. Figure 8 shows the structure of the MPC-based steering control system.

As illustrated in the figure, the MPC-based steering control system consists of error calculation models for the driver model and steering model, the MPC-based driver model, the MPC-based steering model, and the crane model. The steering map shows the relationship between the steering angles at the first and second axles. The crane model was designed using a full three-dimensional (3D) multi-body model in the ADAMS environment, and was used to evaluate the steering and dynamic performances.

In the case of the MPC-based driver model, the value of $e_{d\_p}$ from equation (13) is used in the MPC algorithm to derive the optimal value of ${\delta }_{c,1}$ . In the case of the MPC-based steering model, the value of $e_{d\_f}$ from equation (15) is used in the MPC algorithm to derive the optimal values of ${\delta }_{c,3},{\delta }_{c,4},\mathrm{and}{\delta }_{c,5}$ at the third, fourth, and fifth axles. This study was performed based on the premise that preview information could be acquired based on road information.

The purpose of the above control strategy is to reduce the steering effort by minimizing the driver’s steering input. The control algorithm was designed in the Matlab/Simulink environment.

MPC technique

The MPC algorithm derives the current optimal input based on prediction. This algorithm performs a prediction by regarding the current state as an initial plant value within the finite prediction time horizon, which is used to solve an optimal problem of the finite horizon open-loop system. Unlike the existing control methods that apply a pre-calculated result, the MPC algorithm derives a real-time optimal solution online. This optimal solution can be derived by applying the constraint to every input and state. ¹⁶ Figure 9 shows the concept outline of an MPC.

Figure 9.

MPC concept.

Using these characteristics of MPC, this study develops a control algorithm to obtain the optimal value of ${\delta }_{c,1}$ based on the driver model, and the optimal values of ${\delta }_{c,3},{\delta }_{c,4},\mathrm{and}{\delta }_{c,5}$ in the steering system. The development process of each MPC-based control algorithm is as follows.

MPC formulation for obtaining an optimal value of ${\delta }_{c,1}$ based on the driver model

The objective function of the MPC algorithm to derive the scalar optimal value of ${\delta }_1$ is defined below

$$\underset{{\overrightarrow{\delta }}_{c,1}}{min}{J}_d={\overrightarrow{y}}_{\mathrm{pd}}^T{Q}_{\mathrm{md}}{\overrightarrow{y}}_{\mathrm{pd}}+{{\overrightarrow{\delta }}_{c,1}}^T{R}_{\mathrm{md}}{\overrightarrow{\delta }}_{c,1}$$

(16)

$$\begin{array}{c}{\overrightarrow{y}}_{\mathrm{pd}}={\left[\begin{array}{cccc}{y}_d(k+1)\hfill & y_d(k+2)\hfill & \cdots \hfill & y_d(k+N_p)\hfill \end{array}\right]}^T\hfill \\ {\overrightarrow{\delta }}_{c,1}={\left[\begin{array}{cccc}{\delta }_{c,1}(k+1)\hfill & {\delta }_{c,1}(k+2)\hfill & \cdots \hfill & {\delta }_{c,1}(k+N_p)\hfill \end{array}\right]}^T\hfill \\ Q_{\mathrm{md}}=\mathrm{diag}{\left[\begin{array}{ccc}{\overline{Q}}_{\mathrm{md}}(k+1)\hfill & \cdots \hfill & {\overline{Q}}_{\mathrm{md}}(k+N_p)\hfill \end{array}\right]}^T\hfill \\ R_{\mathrm{md}}=\mathrm{diag}{\left[\begin{array}{ccc}{\overline{R}}_{\mathrm{md}}(k+1)\hfill & \cdots \hfill & {\overline{R}}_{\mathrm{md}}(k+N_p)\hfill \end{array}\right]}^T\hfill \end{array}$$

(17)

where ${\overrightarrow{y}}_{\mathrm{pd}}$ is the output predictor matrix for $y_d$ in equation (17) and ${\overrightarrow{\delta }}_{c,1}$ is the predicted input vector that can be calculated by a Quadratic Programming (QP) algorithm to minimize the objective function J_d. The terms $Q_{\mathrm{md}}$ and $R_{\mathrm{md}}$ in the objective function J_d represent the weighting matrices defined for the relatively different minimization level of each state and input. Through $N_p$ step-ahead prediction (from the current step) with the discrete-time error dynamics, the following output predictor $(y_d)$ can be derived as follows (equation (18))

$$y_d=M_{\mathrm{pd}}{e}_{d\_p}\left(k\right)+H_{\mathrm{pd}}{\overrightarrow{\delta }}_1+F_{\mathrm{pd}}{\delta }_1\left(k\right)$$

(18)

where the matrices $M_{\mathrm{pd}}$ , $H_{\mathrm{pd}}$ , and $F_{\mathrm{pd}}$ are constructed by using $A_{e,d}$ and $B_{e,d}$ from equation (14). The terms $e_{d\_p}(k)$ and ${\delta }_{c,1}(k)$ are the error defined in equation (14) and the steering angle at the current step, respectively. The constructed matrices are given as follows

$M_{\mathrm{pd}}=\left[\begin{array}{c}{A}_{e,d}\hfill \\ \multicolumn{1}{c}{A_{e,d}^2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{A_{e,d}^N}\end{array}\right]$ , $H_{\mathrm{pd}}=\left[\begin{array}{cccc}0\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{B_{e,d}}& 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{A_{e,d}}^{N-2}{B}_{e,d}}& {A_{e,d}}^{N-3}{B}_{e,d}\hfill & \cdots \hfill & 0\hfill \end{array}\right]$ , $F_{\mathrm{pd}}=\left[\begin{array}{c}{B}_{e,d}\hfill \\ \multicolumn{1}{c}{A_{e,d}{B}_{e,d}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{A_{e,d}}^{N-1}{B}_{e,d}}\end{array}\right]$

By rearranging equation (16) with equation (18), the quadratic objective function can be expressed as shown in equation (19)

$$\begin{array}{cc}\multicolumn{1}{c}{J_d}& =\frac{1}{2}{{\overrightarrow{\delta }}_{c,1}}^T(H_{\mathrm{pd}}^T{Q}_{\mathrm{md}}{H}_{\mathrm{pd}}+R_{\mathrm{md}}){\overrightarrow{\delta }}_{c,1}\hfill \\ \multicolumn{1}{c}{}& +({e_{d\_p}}^T\left(k\right)M_{\mathrm{pd}}^T{Q}_{\mathrm{md}}{e}_{d\_p}\left(k\right)+{\delta }_{c,1}\left(k\right)F_{\mathrm{pd}}^T{Q}_{\mathrm{md}}{H}_{\mathrm{pd}}){\overrightarrow{\delta }}_{c,1}\hfill \end{array}$$

(19)

The optimal solution ${\overrightarrow{\delta }}_{\mathrm{opt},c,1}$ in equation (18) minimizes $J_d$ in terms of $e_{d\_p}$ that can be calculated through the QP provided from MATLAB with a set of lower and upper bounds for the steering angle ${\delta }_{c,1}$ at the first axle. The weighting matrix that constitutes the matrices $M_p$ , $H_p$ , and $F_p$ is shown below

$$\mathrm{weighting}\mathrm{matrix}=\left[\begin{array}{cc}a\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& b\hfill \end{array}\right]$$

(20)

where a and b determine the decrease ratio of $e_{1,d}$ and $e_{2,d}$ that are defined in equation (13). For example, if a is larger than b, reducing $e_{1,d}$ will be more focused when driving, and if b is larger than a, the driving is more focused on reducing $e_{2,d}$ relatively. Therefore, elements a and b of the weighting matrix reflect the driving characteristics of a specific driver. The relation between a and b can be expressed as below

$$\frac{b}{a}=m\tanh \left\{n(t\left(j\right)-o)\right\}+p$$

(21)

where $t(j)\in [t_0,t_0+\mathrm{dt},...,t_p-\mathrm{dt},t_p]$ is the preview window, t₀ is the current time, and t_p is the prediction time. In this study, the preview time is defined as 1.5 s and the discretized time step $\mathrm{\Delta }t$ is defined as 0.1. And the number of prediction steps is 20 for the MPC. The terms m, n, o, and p determine b/a. This study used fixed values for the above weighting matrix, preview time, and the update frequency of the steering input to assume that the driving characteristics of a driver are consistent when evaluating the performance of the steering system.

MPC formulation for obtaining an optimal values of $e_{1,d}$ for the steering system

The MPC algorithm to derive the optimal values of ${\delta }_{c,3},{\delta }_{c,4},\mathrm{and}{\delta }_{c,5}$ is similar to the MPC algorithm to derive ${\delta }_{c,1}$ which can be arranged as below

$$\underset{{\overrightarrow{\delta }}_c}{min}{J}_s={\overrightarrow{y}}_{\mathrm{ps}}^T{Q}_{\mathrm{ms}}{\overrightarrow{y}}_{\mathrm{ps}}+{{\overrightarrow{\delta }}_c}^T{R}_{\mathrm{ms}}{\overrightarrow{\delta }}_c$$

(22)

where ${\delta }_c$ is defined in equation (6)

$$\begin{array}{c}{\overrightarrow{y}}_{\mathrm{ps}}={\left[\begin{array}{cccc}{y}_s(k+1)\hfill & y_s(k+2)\hfill & \cdots \hfill & y_s(k+N_p)\hfill \end{array}\right]}^T\hfill \\ {\overrightarrow{\delta }}_c={\left[\begin{array}{cccc}{\delta }_c(k+1)\hfill & {\delta }_c(k+2)\hfill & \cdots \hfill & {\delta }_c(k+N_p)\hfill \end{array}\right]}^T\hfill \\ Q_{\mathrm{ms}}=\mathrm{diag}{\left[\begin{array}{ccc}{\overline{Q}}_{\mathrm{ms}}(k+1)\hfill & \cdots \hfill & {\overline{Q}}_{\mathrm{ms}}(k+N_p)\hfill \end{array}\right]}^T\hfill \\ R_{\mathrm{ms}}=\mathrm{diag}{\left[\begin{array}{ccc}{\overline{R}}_{\mathrm{ms}}(k+1)\hfill & \cdots \hfill & {\overline{R}}_{\mathrm{ms}}(k+N_p)\hfill \end{array}\right]}^T\hfill \end{array}$$

(23)

where ${\overrightarrow{y}}_{\mathrm{ps}}$ is the output predictor matrix for $y_s$ and ${\overrightarrow{\delta }}_c$ is the predicted input vector based on ${\delta }_c$ in equation (5). The terms $Q_{\mathrm{ms}}$ and $R_{\mathrm{ms}}$ are the weighting matrices for $J_s$ . The output predictor $y_s$ is given as follows

$$y_s=M_{\mathrm{ps}}{e}_{d\_f}\left(k\right)+H_{\mathrm{ps}}{\overrightarrow{\delta }}_c+F_{\mathrm{ps}}{\delta }_c\left(k\right)$$

(24)

where the matrices $M_{\mathrm{ps}}=\left[\begin{array}{c}{A}_e\hfill \\ \multicolumn{1}{c}{A_e^2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{A_e^N}\end{array}\right]$ , $H_{\mathrm{ps}}=\left[\begin{array}{cccc}0\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{B_e}& 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{A_e}^{N-2}{B}_e}& {A_e}^{N-3}{B}_e\hfill & \cdots \hfill & 0\hfill \end{array}\right]$ , and $F_{\mathrm{ps}}=\left[\begin{array}{c}{B}_e\hfill \\ \multicolumn{1}{c}{A_e{B}_e}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{A_e}^{N-1}{B}_e}\end{array}\right]$ are determined by $A_e$ and $B_e$ from equation (11). The terms $e_{d\_f}(k)$ and ${\delta }_c(k)$ are the error in equation (15) and the steering angle at the current step, respectively. By rearranging equation (22) with equation (24), the quadratic objective function can be expressed as shown in equation (25)

$$\begin{array}{cc}\multicolumn{1}{c}{J_s}& =\frac{1}{2}{{\overrightarrow{\delta }}_c}^T(H_{\mathrm{ps}}^T{Q}_{\mathrm{ms}}{H}_{\mathrm{ps}}+R_{\mathrm{ms}}){\overrightarrow{\delta }}_c\hfill \\ \multicolumn{1}{c}{}& +({e_{d\_f}}^T\left(k\right)M_{\mathrm{ps}}^T{Q}_{\mathrm{ms}}{e}_{d\_f}\left(k\right)+{\delta }_c\left(k\right)F_{\mathrm{ps}}^T{Q}_{\mathrm{ms}}{H}_{\mathrm{ps}}){\overrightarrow{\delta }}_c\hfill \end{array}$$

(25)

The optimal solution ${\overrightarrow{\delta }}_{\mathrm{opt},c}$ minimizes the objective function $J_s$ with respect to $e_{d\_f}(k)$ by considering a set of lower and upper bounds for the steering angle at each axle. ¹⁷ Using this ${\overrightarrow{\delta }}_{\mathrm{opt},c}$ , the optimal steering angles for the steering system at the third to fifth axles $({\delta }_{c,3},{\delta }_{c,4},{\delta }_{c,5})$ can be obtained. The additional restriction for QP, that is, the relationship between the first and second steering angles (Figure 1), is also considered.

Evaluation of steering performance

The performance of the developed steering control algorithm was evaluated in the ADAMS environment. A double lane change and a curved path (radius of gyration: 90 m), which frequently occur in road-steering mode, were selected as the scenarios for evaluation. Figure 10 shows each selected scenario.

Figure 10.

Double lane change (left) and curved path scenario (right).

Simulations were performed with the driving speed conditions of 28, 50, 72 km/h, which represent three speed ranges in road-steering mode (see Figure 2). Based on the analysis results, the steering performances of both the proposed steering control system and the conventional steering system were compared and analyzed.

As conditions for the simulation, the preview time $t_p$ of the driver model was set to 1.5 s, the discretized time $t_d$ to 0.1 s, the prediction step $N_p$ to 20 steps, and the up-date cycle of the driver’s steering input to 0.1 s. As a criterion for the steering performance evaluation, the driver’s steering effort was defined as an amount of the steering input and its equation is given below

$$S.E=\underset{o}{\overset{t}{\int }}\left|{\stackrel{·}{\delta }}_{c,1}\left(t\right)\right|\mathrm{dt}$$

(26)

where t is the simulation time and ${\stackrel{·}{\delta }}_{c,1}$ is the steering angle rate at the first axle computed from the human driver model. If the steering effort has a high value due to difference between dynamic responses that driver intended and actual dynamic responses of the vehicle while driving, the driver fatigue will be grown rapidly and it can cause fatal accidents. The steering effort (S.E.) represents the total amount of absolute steering angle at the first axle that is generated in the considered scenarios during the simulation time. Since the steering angle at the first axle is set directly by the driver through the hydraulic power steering, we call steering effort the “driver’s steering effort” in the study.

Simulation setup in ADAMS environment

The steering control algorithm proposed in this study uses a human driver model and a MPC algorithm to derive the optimal steering angle.

A 3D multi-body crane model modeling in ADAMS environment

A typical mechanical system consists of many components. ADAMS is a multi-body dynamics analysis software that simulates and analyzes the dynamic behavior of mechanical systems according to the time change. The information regarding the behavior of components connected with joints under kinematic conditions of large displacement, including not only the position of each component according to the time change, velocity, acceleration, and joint reaction but also the natural frequency of the system, mode shape, and state-space matrix, can be acquired through a multi-body dynamic analysis using ADAMS. Moreover, kinematic, static, and quasi-static analyses of a mechanical system, as well as linear, non-linear, and dynamic analyses, are all possible.

Based on the CATIA design of the mobile crane to be developed, a dynamic model was configured for the main components related to the vehicle dynamics by using the ADAMS. The configured dynamics model can be largely divided into two parts: the top and bottom structure of the vehicle. The vehicle’s bottom part consists of each steering axle, the engine, the body, the tires, and so on. Because most of the components of the bottom structure can affect the vehicle’s steering performance, the dynamic model was configured using an actual model. The top structure of the vehicle was configured using a concentrated mass (Figure 11).

Figure 11.

Dynamic model of all-terrain crane.

Figure 12 illustrates a schematic diagram of the co-simulation in which the Matlab/Simulink-based control system and the ADAMS-based multi-body dynamics analysis model of the crane are linked together for their simultaneous operation like a single system. While the yaw rate, position, and velocity were set as the output variables of the ADAMS model in the figure, the steering angles for each axle was set as the input variables of the ADAMS model that are computed from the Matlab/Simulink model. The x and y positions of the center of mass, yaw angle, longitudinal velocity, and lateral velocity of the crane are sent to Matlab/Simulink every 0.01 s. The steering angle of each axle is then calculated at 0.01 s intervals using the transmitted data in the Matlab/Simulink-based control system, and is sent back to ADAMS.

Figure 12.

Co-simulation between ADAMS and Simulink models.

Simulation results

Case 1: double lane change scenario

In the double lane change scenario, the simulation conditions for the driving time and driving distance per driving speeds are shown in Table 1.

Table 1.

Conditions for driving time and driving distance per driving speeds in double lane change scenario.

Driving speed (km/h)	Driving time (s)Conventional	Driving time (s)MPC-based	Driving distance (m)
28	174.92	173.80	1300
50	92.12	92.14	1200
72	87.68	87.72	1600

MPC: model predictive control.

Figures 13 and 14 and Tables 2 and 3 show the simulation results of the steering angle and steering effort.

Figure 13.

Steering angle and steering effort in double lane change: (a) velocity: 28 km/h, (b) velocity: 50 km/h, and (c) velocity: 72 km/h.

Figure 14.

Dynamic behavior in double lane change: (a) velocity: 28 km/h, (b) velocity: 50 km/h, and (c) velocity: 72 km/h.

Table 2.

Steering effort comparison in double lane change scenario.

Steering effort (steering)		Value (°)		Rate of change
		Convention system	MPC-based system
28 km/h	S.E.	33.05	14.48	−56.19%
50 km/h	S.E.	10.68	4.20	−60.67%
72 km/h	S.E.	5.53	1.55	−71.97%

MPC: model predictive control.

Table 3.

Dynamic behavior comparison (angle (°), rate (°/s)).

Velocity (km/h)	Yaw rate (MAX)		Roll angle (MAX)		Roll rate (MAX)
	Con	MPC	Con	MPC	Con	MPC
28	7.7	2.02	0.52	0.36	0.99	0.80
50	4.22	2.35	0.46	0.41	0.69	0.77
72	2.41	1.12	0.35	0.38	0.42	0.63

MPC: model predictive control; Con: convention system.

The simulation result shows that in the double lane change scenario, the driver’s steering input of the MPC-based steering system can be significantly reduced by steering assistance at the third, fourth, and fifth axles. Moreover, although the dynamic states (yaw rate, roll angle, and roll rate) of the crane in the MPC-based steering system can be stabilized compared to the existing system, the roll angle and rate at 72 km/h, as well as the roll rate at 50 km/h, have relatively large values. This is because the steering control system is not optimized with consideration of the driver’s driving intention (i.e. various driving characteristics for individual drivers). Nevertheless, it was confirmed that the proposed MPC-based steering system efficiently reduced the driver’s steering effort compared to the existing system on a comprehensive level, and that its dynamic stability could be improved in all conditions except for the roll angle and rate at 72 km/h, as well as the roll rate at 50 km/h.

Case 2: curved path scenario (R = 90 m)

In the curved-path scenario with the radius of gyration being 90 m, the simulation conditions for the driving time and driving distance per driving speeds are shown in Table 4.

Table 4.

Conditions for driving time and driving distance per driving speeds in curved-path scenario with radius of gyration of 90 m.

Driving speed (km/h)	Driving time (s)Conventional	Driving time (s)MPC-based	Driving distance (m)
28	96.37	97.44	730
50	69.03	69.35	730
72	52.57	52.77	730

MPC: model predictive control.

Figures 15 and 16 and Tables 5 and 6 show the simulation results.

Figure 15.

Steering angle and effort in curved path: (a) velocity: 28 km/h, (b) velocity: 50 km/h, and (c) velocity: 72 km/h.

Figure 16.

Dynamic behavior in curved path: (a) velocity: 28 km/h, (b) velocity: 50 km/h, and (c) velocity: 72 km/h.

Table 5.

Steering effort comparison in curved path scenario.

Steering effort (steering)		Value (°)		Rate of change
		Convention system	MPC-based system
28 km/h	S.E.	13.25	6.66	−49.74%
50 km/h	S.E.	14.47	12.26	−15.27%
72 km/h	S.E.	30.62	8.65	−71.75%

MPC: model predictive control.

Table 6.

Dynamic behavior comparison (angle (°), rate (°/s)).

Velocity (km/h)	Yaw rate (MAX)		Roll angle (MAX)		Roll rate (MAX)
	Con	MPC	Con	MPC	Con	MPC
28	5.31	6.12	0.34	0.53	0.36	0.45
50	9.23	9.24	1.06	1.11	1.16	1.10
72	14.26	12.82	2.20	2.03	2.06	2.22

MPC: model predictive control; Con: convention system.

The MPC-based steering system proposed in the curved-path scenario can effectively reduce the driver’s steering effort. However, the yaw rate was relatively high at speeds of 50 km/h or less, and the values for roll rate and angle were slightly higher in the MPC-based system except for the roll angle at 72 km/h and the roll rate at 50 km/h, although no significant difference was observed between the two systems. On a comprehensive level, the proposed MPC-based steering control system revealed a better performance compared to the existing steering system, as the proposed MPC-based steering system could effectively reduce the driver’s steering effort, and showed the dynamic stabilization almost equivalent to that of the existing steering system at the same time. Considering the different driving characteristics among drivers, however, it can be determined that an optimized control algorithm must be applied in order to reasonably reflect the driving intentions of the driver.

Conclusion

The purpose of this study is to develop an optimal steering control algorithm that can reduce a driver’s steering effort for an all-terrain crane, and to evaluate the performance of the developed algorithm with a human driver model. A bicycle model was used to derive a simplified model for a crane steering system. A MPC-based control algorithm was applied to derive the optimal steering angle for each axle. The performance of the control algorithm developed in the Matlab/Simulink environment was evaluated with the human driver model through co-simulation with the crane dynamics model developed in the ADAMS environment. The performance verification was carried out for double lane change and curved-path scenarios at various speeds (28, 50, and 72 km/h) in road-steering mode.

In the co-simulation configuration where the ADAMS-based dynamics model and Matlab/Simulink-based control system were interconnected, the vehicle’s position, velocity, and yaw information were set as the output variables for the ADAMS model. Moreover, the steering angle of each axle was also set as an input variable for the ADAMS model that was calculated by the MPC-based control algorithm with the Matlab/Simulink. Here, the time step for co-simulation was set to 0.01 s.

The analysis results confirmed that the MPC-based steering system effectively reduced the driver’s steering effort compared to the existing steering system, and that the dynamic stability (yaw rate, roll angle, and roll rate) of the existing steering system for a crane can be secured in the proposed system. However, because the driving characteristics are different for individual drivers, the development of an optimized steering control system reflecting specific driving characteristic (e.g. steering angle rate by applying inequality constraints to the MPC) is considered as future work. Since driver’s intention was not considered in the proposed steering control algorithm, the driving stability of a crane may not be secured in double lane change scenario due to conflicts between control inputs and driver’s inputs in a practical system. Therefore, the proposed steering control algorithms can be extensively developed by incorporating the driver’s driving intention (i.e. driver’s unique driving characteristics) as a future work, which can improve the performance of both vehicle stability and steering controls through predictions of driver’s input with the MPC algorithm. For example, the MPC algorithm may compute optimal steering inputs by predicting the driver’s inputs based on the driver’s intention. Furthermore, it is planned as future works to develop multiple constraints of the MPC for improving the yaw and roll stability of the crane while reducing steering efforts because the steering control algorithm proposed in this study does not consider stability.

In this study, it was assumed that the desired yaw rate is close to zero. However, the desired yaw rate is not close to zero if the crane is driving on a winding road. Therefore, the MPC technique needs to be improved by including an additional component that enables to deal with non-zero desired yaw values and thus to enhance the steering control performance.

The proposed MPC-based steering control algorithm for reducing a driver’s steering effort can be applied extensively to various multi-axle vehicles including cranes. The steering control algorithm developed in this study can be implemented in an actual crane using steering angle sensors (encoders) and cameras to improve driving stability and steering efficiency. It is expected that the simulation and analysis techniques developed in this study can also be utilized for the design phase.