Trajectory tracking and traction coordinating controller design for lunar rover based on dynamics and kinematics analysis

2.1. Wheel-soil interaction model

In order to research the wheel-soil interaction, a single wheel is well choosen. We assume that the entire wheel is stiff relative to the ground so that we can consider the wheel is rigid. Fig. 1 shows the forces and torques acting on the single wheel of a lunar rover [11]. Based on the terramechanics model, the contact forces and torques exerting on the wheels are calculated and optimized in this paper. In Fig. 1, $F_x$, $F_y$ are the forces exerted on the wheel by rocker or bogie joint, $N$ is the normal force applied to the wheel, $T$ is the traction torque applied to the wheel, $F_H$ is the soil thrust acting on the wheel, $F_R$ is the motion-resistance force applied to the wheel, $r$ is the radius of the wheel, $Z$ is the penetration depth. Assuming that ${\mu }_0$ is the static friction coefficient, $\mu$ is the dynamic friction coefficient. When the wheel purely rolls, the force acts on wheel is static friction force with the value is $F_{static}={\mu }_{0}N$. When the wheel has a relative motion with the ground (with slip phenomenon), friction force becomes $F_{dynamic}=\mu N$.

For each wheel, the net longitudinal force $F_{DP}$ is the difference between the soil thrust $F_H$ and the motion-resistance force $F_R$. The maximum net longitudinal force can be calculated as follows:

For the motion-resistance force $F_R$, which consists of the resistance of terrain $F_b$ and the compaction resistance $F_c$, it can be calculated as follows:

As the traction forces given to the ground are limited by wheel and the adhesion condition of the ground under real conditions, $F_{DP}\le {\mu }_{0}N$ should be meet at all times.

The rolling resistance torque or friction torque $T_R$ is as follows:

where $r$ is the radius of the wheel.

Fig. 1Forces and torques acting on a rigid wheel

2.2. Longitudinal dynamics model

The rover is prone to slip when traverses over soft or challenging terrain. If the differences among slip ratios of each wheel are too large, the power consumption will increase and dangerous situations occurring. So it is necessary to coordinate and reduce each wheel’s slip ratio, and can effectively improve the obstacle-climbing capability and mobility. The slip ratio is defined as follows:

where $w$ is the angular speed of wheel, $v$ is the traveling velocity of wheel center.

In this definition, the value of $\lambda$ is between –1 and 1. When the wheel speed $r_w$ is greater than the velocity of wheel center $v$, $\lambda$ is positive. Conversely, when the wheel speed $r_w$ is less than the velocity of wheel center $v$, $\lambda$ is negative. When $\lambda$ is zero, wheels roll purely.

The following dynamical equations can be listed from the force analysis [12] for the rover model (bilateral symmetry) only considering the motion of longitudinal direction:

where $m_0=m/2$, $m$ is the mass of the whole body, $F_{Hi}$ is the soil thrust of the wheel $i$, $F_{Ri}$ is the motion-resistance force of the wheel $i$, $I_w$ is the wheel inertia, $T_i$ is the wheel traction torque of wheel $i$, $T_{Ri}$ is the rolling resistance torque of wheel $i$.

Bring the Eq. (5) and (6) into the derivation of the Eq. (4), we can obtain the following equation:

where:

2.3. Trajectory tracking control model

Six wheeled lunar rover with rocker bogie consists of six driven wheels (front and rear four wheels with steering function), suspension system and body. Lunar rover is driven by the same mechanism with left and right independent. We combine differential steering with independent steering to achieve steering of lunar rover. The longitudinal slip is considered when discussing the rover kinematics. Assumptions are made as follows: 1, the lunar rover is all rigid without any flexible body involving. 2, steering axis of each steering wheel is perpendicular to the ground. 3, the orientation error $e_{\theta }$ generated during the vehicle driving is minor.

Fig. 2The diagram between kinematics path and lunar rover

A coordinate system based on path is established as shown in Fig. 2. The centroid position of the rover is $A$; the orientation error $e_{\theta }$ is the angle between forward direction of lunar rover and the positive $X$ axis; the speed error $\mathrm{\Delta }v$ is the velocity difference between left and right wheel; the lateral error $e_d$ is the coordinate of centroid point in the coordinate system.

The angles of each steering wheel are calculated as follows for a specified turning radius $R$:

where $l$ is the front and rear wheelbase, $d$ is the width of the body, $R$ is the instantaneous turning radius of body movement, $\alpha$ is steering azimuth, ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, ${\delta }_4$ are left front, left rear, right front and right rear wheel steering angle respectively.

According to the above four equations, steering angles of each steering wheel can be calculated when tracking a trajectory with fixed radius. At the beginning of tracking trajectory, the angles of each steering wheel should be adjusted to the corresponding values. During the process of tracking, the errors will inevitably occur because of the rough terrain. In order to protect the mechanism to avoid damage by frequent swing of steering wheels, we can achieve fine-tune of the orientation error and lateral error by means of left and right wheel differential steering when the desired curvature is constant, then the kinematics model of lunar rover can be achieved as shown in follows:

where $v_l$ and $v_r$ are the left and right average speeds of lunar rover respectively. $e_{\theta }$ is assumed to be minor, so $\mathrm{s}\mathrm{i}\mathrm{n}{e}_{\theta }\approx e_{\theta }$.

The torque $T$ is chosen as input signal, while the output signal is the acceleration of wheel. Based on least squares method, the driven system can be identified and the transfer function is as follows:

From Eqs. (12), (13) and (14), the state-space of trajectory tracking control model can be derived:

where $x={\left[\mathrm{\Delta }\dot{v}\mathrm{}\mathrm{}\mathrm{\Delta }v{e}_d{e}_{\theta }\right]}^T$ is state variable, $\mathrm{\Delta }u$ is the system input (input torque $T$). The state equation is:

where $C=\left[0\mathrm{}0\mathrm{}1\mathrm{}0\right]$, $D=\left[0\right]$.

3.1. Design of traction coordinating controller

The traction coordinating controller designed in this paper is shown in Fig. 3, where $V_d$ is the desired speed (about 200 m/h), $V_{cheti}$ is the actual speed. We can regulate the lunar rover to track the desired speed by inputting the difference between $V_d$ and $V_{cheti}$ to the PID controller, also the total demand torque $T_v$ can be calculated. The torque $T_v$ should be allocated to the six wheels reasonably. The maximum traction force a terrain can bear increases with increasing normal force, so we set the coefficient $k_n$ ($i=$1-6, $k_{v1}+k_{v2}+\dots +k_{v6}=$1) to make up the disadvantage that sliding mode controller neglects the influence of normal forces on the traction torque. $T_{vi}$ which meets the equation $T_{vi}=k_{vi}{T}_v$ is the torque applied to wheel $i$. The ratios of $T_{vi}$ to the normal force of every wheel are equal, the greater normal force, the greater traction torque exerted on the wheel, thereby the traction force a terrain can bear increases to enhance the driving efficiency. Then adjusting the $T_{vi}$ with the control output $u_i$ of sliding mode controller, to control the slip ratio of every wheel and gain the output $U_i$ of traction coordinating control.

The slip ratio of each wheel can be controlled effectively using the traction coordinating controller on the basis of controlling the velocity of lunar rover, which can reduce the force consumption, enhance the trafficability and improve the traction coordinating performance. According to the nonlinear characteristics of the longitudinal dynamics model of lunar rover, it is suitable to select the improved sliding mode controller to control the slip ratio. Taking the average slip ratio ${\stackrel{-}{\lambda }}_i$ as the control target with using the improved approach law. Assuming the state error is $e_i={\lambda }_i-{\stackrel{-}{\lambda }}_i$, the switching function is as follows:

Fig. 3Diagram of traction coordinating control system

If the desired sliding mode control is achieved, the equivalent control method is defined as $s\&=0$, combining Eq. (7) and $s\&=0$ results in:

In order to meet the reaching conditions of sliding mode control, reach the sliding surface in the shortest time and ensure that the system has good robustness, while trying to weaken the chattering, this paper adopts the improved exponential approach law [13, 14]:

where $\mathrm{s}\mathrm{g}\mathrm{n}\left(\right)$ is the sign function. Substitution the derivation of Eq. (7) into Eq. (18) gives:

Combining Eq. (19) and (20), we obtain designing result of controller as follows:

where $r_{1i}$ and $r_{2i}$ are both weight coefficients, $\epsilon >$0 and $k>$ 0 are the control parameters. In order to ensure reduced buffeting and quickly approaching, $\epsilon$ should be reduced while increasing the value of $k$, and the sign function also should be replaced by the saturation function. As:

The proposed law meets the reaching condition ($s\dot{s}<0$) of the sliding mode control. Then, the entire movement of the system is asymptotic stability.

3.2. The design of trajectory tracking controller

Under the precondition of wheels’ traction coordinating control, the corresponding trajectory tracking controller should be designed to track the desired trajectory of the lunar rover. The optimal control algorithm can be used to achieve trajectory tracking because the open-loop system of Eq. (16) is observability and controllability [15, 16].

The rover system can be regarded as a linear system proximately under the certain constraint condition. So the optimal problem of this system is quadratic optimal control problem with a purpose of seeking an optimal control signal $u\left(t\right)$, to make the evaluation index function minimum:

where $Q$ and $R$ are both weighted matrices.

The optimal state feedback array $K$ can be obtained by solving the Riccati algebraic equation. That is to say:

is the control quantity of trajectory tracking, $K_1$, $K_2$, $K_3$, $K_4$ are both state reaction coefficients.

The control quantity of trajectory tracking should be exerted on each wheel on the basis of traction coordinating control to correct the lateral error and the orientation error, a deviation-rectifying torque $\mathrm{\Delta }u/2$ is added or subtracted to the control quantity of sliding mode traction coordinating control for each wheel in this paper. The deviation-rectifying quantities of four wheels (wheel 1, 2, 5, 6) are exerted according to the following proportion. Taking the turning for example, we assume $R$ is the turning radius, the rover moving around the turning center in a clockwise direction, as shown in Fig. 2. The torque of each wheel except the four turning wheels’ angles is divided as follows:

where:

and $R$ is the turning radius of lunar rover, $d$ is the width of lunar rover, $l$ is the length of lunar rover.

Based on the above theory, the trajectory tracking control is achieved considering the wheel slip in this paper, namely the trajectory tracking control algorithm with slip compensation. The control block diagram of the proposed algorithm is shown in Fig. 4.