7. Rear Axle Control Strategies

In a 4-Wheel Steering (4WS) agricultural robot, the rear axle is not merely a follower; it is an active participant in path tracking. Its primary roles are: * Turning Radius Reduction: By steering in the opposite direction to the front wheels (counter-phase), the robot can negotiate tight headland turns. * Crab Steering: By steering in the same direction as the front wheels (in-phase), the robot can compensate for lateral sliding on slopes, maintaining the chassis parallel to the path. * Heading Stabilization: The rear axle provides an additional degree of freedom to decouple the regulation of the lateral error (\(y\)) from the heading error (\(\tilde{\theta}\)), leading to much smoother trajectories.

7.1. Curvature-Based Control Law

The control strategy for the rear axle implemented in this simulator is based on an analytical approach that relates the steering effort to the path's geometry.

7.1.1. Quadratic Resolution

The core logic aims to satisfy a specific heading error convergence. Unlike traditional controllers, we solve for the steering angle that satisfies a second-order geometric relationship.

Let us define the desired variation of the heading error as a proportional correction:

\[ \Delta\tilde{\theta} = K_{d2} \cdot (\tilde{\theta} - \tilde{\theta}_{desired}) \]

Where \(K_{d2}\) is a tuning gain and \(\tilde{\theta}_{desired}\) is the target heading (usually zero).

The simulator seeks to solve the following quadratic equation for a virtual term \(X\):

\[ X^2 - K_d \cdot X + c(s) \cdot \Delta\tilde{\theta} = 0 \]

Where \(K_d\) is a damping-like gain and \(c(s)\) is the local curvature. To find the solution, we calculate the discriminant (\(\Delta\)):

\[ \Delta = K_d^2 - 4 \cdot c(s) \cdot \Delta\tilde{\theta} \]

This discriminant is a critical stability indicator: 1. If \(\Delta \ge 0\): Real solutions exist. We select the root that minimizes the steering effort:

\[ RootTerm = \frac{K_d - \sqrt{\Delta}}{2} \]

If \(\Delta < 0\): The required correction is mathematically impossible for the current curvature/error state (complex solutions). The simulator triggers a safety warning and clips \(\Delta\) to \(0\) to maintain numerical continuity.

The target effective orientation for the rear axle is then derived from the ratio of this root to the curvature:

\[ \tan(\tilde{\theta} + \beta_R + \delta_R) = \frac{Root\_Term}{c(s)} \]

A major mathematical challenge arises when the robot moves in a straight line. As the curvature \(c(s)\) approaches zero, the term \(\frac{RootTerm}{c(s)}\) tends toward an undefined form (\(\frac{0}{0}\)).

To ensure a seamless transition, the simulator detects when \(|c(s)| < 0.01\). In this case, it abandons the quadratic resolution and switches to a Linear Term derived from a first-order Taylor expansion. This term reintroduces the lateral error (\(y\)) to ensure the rear axle helps the robot "stick" to the line:

\[ LinearTerm = \frac{\Delta\tilde{\theta}}{K_d} - \frac{K_d \cdot y}{4} \]

By using this logic, the robot maintains high-precision tracking on straight segments where the quadratic equation would otherwise become singular.

7.1.3. Dynamic Compensation and Saturation

Once the target tangent (either from the quadratic root or the linear term) is determined, we must solve for the actual mechanical steering angle \(\delta_R\).

Heading and Slip Inversion: The angle we calculated is the absolute orientation of the velocity vector. To find the mechanical angle \(\delta_R\) relative to the chassis, we must subtract the current heading error and the estimated dynamic slip angle:

\[ \delta_{R, raw} = \arctan(TargetTangent) - \tilde{\theta} - \beta_R \]

Safety Bounding: To protect the physical integrity of the actuators and prevent tangent singularities (as discussed in section 3.3), a strict saturation is applied:

\[ \delta_R = \max(\delta_{min}, \min(\delta_{max}, \delta_{R, raw})) \]

where \(\delta_{min} = -30^\circ\) and \(\delta_{max} = 30^\circ\).

This final command ensures that the rear axle actively corrects the robot's posture while accounting for the non-linear physics of tire slip and path curvature.

Front_Axle.sce

function Rear_Axle_Output = Model_Rear_Axle_Output(...
Lateral_Error, ...
Heading_Error, ...
Curvature, ...
Front_Beta, ...
Rear_Beta)

// ---------- Parameters ----------
Kd  = 0.4        ;
Kp  = (Kd^2) / 4 ;
Kd2 = 0.5        ;

global Heading_Error_Desired ;
Min_Angle = -30  ;
Max_Angle =  30  ;

// ---------- Computation ----------
Heading_Error_Diff = Kd2 * (Heading_Error - Heading_Error_Desired) ; 
Discriminant = (Kd^2) - 4 * Curvature * Heading_Error_Diff         ;

// If the solution become complexe
if Discriminant < 0 then
    disp("Warning : Compelexe solution !") ;
    Discriminant = 0                       ;
end

Root_Term = 0.5 * (Kd - sqrt(Discriminant)) ; 

// ---------- Singularity for straight line ----------
if abs(Curvature) < 0.01 then
    Linear_Term = (Heading_Error_Diff / Kd) - (Kd * Lateral_Error / 4)       ;
    Rear_Steer_A = atan(Linear_Term) - Heading_Error - Rear_Beta             ;
else
    Rear_Steer_A = atan(Root_Term / Curvature) - Heading_Error - Rear_Beta   ;
end

// ---------- Saturation ----------
if Rear_Steer_A > (Max_Angle * %pi / 180) then
    Rear_Steer_A = Max_Angle * %pi / 180 ;
elseif Rear_Steer_A < (Min_Angle * %pi / 180) then
    Rear_Steer_A = Min_Angle * %pi / 180 ;
end

// ---------- Output ----------
Rear_Axle_Output = Rear_Steer_A ;

endfunction

7. Rear Axle Control Strategies

7.1. Curvature-Based Control Law

7.1.1. Quadratic Resolution

7.1.2. Singularity Management (Straight-Line Navigation)

7.1.3. Dynamic Compensation and Saturation