3.1. Chassis and Suspension System

The design of the chassis of the vehicle resembles the shape of the letter “T” as seen in Fig. 2. It is made of five sheets of aluminum with one for the front, back, left, right, and bottom. Each of these sheets is bent into shape and riveted together to form a rigid chassis. This design features two internal layers which are utilized for the steering and driving systems. As highlighted in Fig. 2, attached to each wheel is an upper and lower control arm which is inspired by a double-wishbone suspension system [Reference Moreno Ramírez, Tomás-Rodríguez and Evangelou37] which traditionally uses two “wish-bone” shaped arms to control the vertical motion of the wheel. The lower control arm is custom-made with water jet steel to ensure durability. Each suspension features gas shocks that are reinforced by coils to improve maneuverability in rough terrain. This kind of system has the advantage of creating negative camber as the wheel travels through its range vertically. Because of this, the vehicle can achieve greater handling abilities due to improved stability as tires can maintain contact with the surface.

3.2. Driving and Steering System

Next, the driving and steering systems are discussed in this section. As previously mentioned, the “T” shape chassis is designed with two internal layers housing the necessary driving and steering components to actuate the vehicle. Each layer has a surface area of approximately 0.10 m². Figure 3 illustrates the driving layer and the steering layers.

Figure 2.

Front view of the SMWV.

Figure 3.

Internal layers of the SMWV.

Starting from the bottom with the driving layer, there are two DC motors per axle for a total of eight independently driven wheels. Each DC motor is attached to a 33:1 gearbox to reduce the total rotational speed while increasing the output torque. More specifically, the maximum rotational output speed is 217 rpm while the nominal output torque is 2.41 Nm. With these specifications, the vehicle can achieve a top speed of approximately 1.80 m/s. Due to the limitations imposed by the dimension of the chassis, it is not possible to align two motors along an axle for a direct drive system unless the motors were placed at an angle. This would create an inefficient drive train; therefore, a belt drive system with a 1:1 ratio per wheel is implemented instead. This setup enables the motors to be mounted in parallel with the axle axis as illustrated in Fig. 4. In this figure, the top and bottom DC motors are driving the left and right wheels, respectively. Placed in between the sidewall of the chassis and the output pulley is an encoder that is mounted on the output shaft.

Figure 4.

Top view of the 1st axle in the driving layer.

The steering layer sits approximately 6.35 cm on top of the driving layer where linear actuators are attached to each wheel assembly through a tie rod. Each actuator has a total stroke of 50 mm with 25 mm being the neutral position. Steering of each wheel is accomplished by extending and retracting each actuator. In Fig. 5, an extended left actuator with a retracted right actuator would steer both wheels to the left. The tightest turn would happen at full extension and retraction, respectively. Built into each actuator are potentiometers that provide stroke position feedback. The maximum stroke of each linear actuator and achievable steering angles are 50 mm and 35 degrees, respectively. The relationship between this feedback and the achieved steering angle is derived experimentally as seen in Fig. 12, in the following sections of the paper.

Figure 5.

Top view of the 1st axle in the steering layer.

3.3. Kinematics Model

3.3.1. Robot Position and Reference Frame

In this section, the kinematics model of the SMWV is derived. Siegwart & Noorkbash’s reference book is utilized to provide a guideline to develop the kinematic model [Reference Siegwart, Nourbakhsh and Scaramuzza38]. The complete kinematic model also utilized attributes from the two-wheel bicycle model of a vehicle. Figure 6 demonstrates the relationship between the global reference frame (signified by axes X and Y) and the local reference frame oriented in the direction of the wheelbase (B) and the longitudinal axis (L).

Figure 6.

Global (X,Y) and Local (L,B) reference frame of SMWV.

3.3.2. Robot Maneuverability

The SMWV is composed of eight standard steerable wheels which increase complexity when analyzing the maneuverability of the vehicle. The first component to analyze is the degree of mobility ( $\delta$ _m) which is a measure of the number of degrees of freedom of the robot chassis that can be immediately manipulated through changes of wheel velocity [Reference Siegwart, Nourbakhsh and Scaramuzza38]. As illustrated by Fig. 7, a simplified bicycle model along the longitudinal axis is illustrated between the physical wheels of the vehicle. More specifically, $\left( {{\delta _{Li}},\;{\delta _{Ri}}} \right)$ denotes the steering angles of the wheels of the vehicle while $\left( {{\boldsymbol\delta _{\bf 1}},\;{\boldsymbol\delta _{\bf 4}}} \right)$ denotes the steering angle average of the first and last axle.

Figure 7.

Kinematic model of 8 × 8.

As seen in Fig. 7 for the complete kinematic model of the vehicle, the eight wheels can be found to connect to an instantaneous centre of rotation which contributes one kinematic constraint. Therefore, the mobility can be calculated as a difference between the number of constraints and total possible degrees of freedom of the SMWV which is equal to 3, thus the degree of mobility ( $\delta$ _m ) is seen to be equal to 2.

Another important parameter is the degree of steerability ( $\delta$ _s ) which considers that steered wheels do not have an instantaneous effect on the pose of the robot chassis but affect the pose over time. The degree of steerability accounts for each independent, steerable wheel. Due to the SMWV having eight steerable wheels, in general, the degree of steerability for this vehicle is 8. However as seen in proceeding sections to streamline vehicle control, several steering configurations which coordinate and limit wheel steering angles have been developed. This, therefore, reduces the degree of steerability but allows for useful steering maneuvers.

Finally, the degree of maneuverability ( $\delta$ _M) can be defined as the overall degrees of freedom that a robot can manipulate. As seen in Eq. (1), the degree of maneuverability ( $\delta$ _M) is the sum of the degree of mobility ( $\delta$ _m) and the degree of steerability ( $\delta$ _s)

(1) \begin{align}{\delta _{\rm{M}}} = {\delta _{\rm{m}}} + {\delta _{\rm{s}}} = 2 + 8 = 10\end{align}

3.3.3. Robot Kinematic Model and Steering Angle Constraints

The linear velocities are calculated by the product of the total longitudinal velocity, $v$ , and the vehicle orientation with respect to the x-axis, $\theta $ , along with the angle between the direction of the velocity with respect to the longitudinal axis of the vehicle, $\varphi $ . The rate of change of the heading angle is denoted by $\dot \theta $ , which is calculated by considering the length to the center of gravity (CG) of the vehicle from the front and rear axles. Equation (2) represents the nonlinear continuous-time relationships of the different velocities of the system where ( $\dot{\textrm{x}} $ , $\dot{\textrm{y}} $ ) are the linear velocities along the respective axis.

(2) \begin{align}\left[ \begin{array}{l} {\dot x} \\[2pt] {\dot y} \\[2pt] {\dot \theta } \end{array} \right] = v*\left[ \begin{array}{l} {\cos (\theta + \varphi )} \\[2pt] {\sin (\theta + \varphi )} \\[3pt] {{\frac{\cos (\varphi )}{{l_1} + {l_4}}}(\tan {\delta _1} - \tan {\delta _4})} \\ \end{array} \right]\end{align}

To find the velocity at the CG, the velocity average of the first and last axle of the vehicle, $\left( {{v_1},\;{v_4}} \right)$ , is calculated as shown below:

(3) \begin{align}\begin{array}{*{20}{c}}{v = \;\dfrac{{{v_1}\cos \left( {{\delta _1}} \right) + {v_4}\cos \left( {{\delta _4}} \right)}}{{2\cos \left( \varphi \right)}}\;}\end{array}\end{align}

Moving forward, the angle between $v$ and the longitudinal axis of the vehicle is calculated with Eq. (4):

(4) \begin{align}\begin{array}{*{20}{c}}{\varphi = {{\tan }^{ - 1}}\left( {\dfrac{{{l_1}\tan \left( {{\delta _1}} \right) + {l_4}\tan \left( {{\delta _4}} \right)}}{{{l_1} + {l_4}}}} \right)}\end{array}\end{align}

To simplify Eqs. (3) and (4), the path curvature of the vehicle, $\sigma $ , along with some assumptions are considered. Starting with the curvature equation which is calculated as the inverse of the turning radius which is better defined as the quotient of angular and linear velocities. This relationship is illustrated in Eq. (5):

(5) \begin{align}\begin{array}{*{20}{c}}{\sigma = {R^{ - 1}} = \dfrac{{\dot \theta + \dot \varphi }}{v}\;}\end{array}\end{align}

Besides the curvature equation, the necessary assumptions made to simplify Eq. (2) include setting the CG location to the middle of the vehicle body and assuming the velocities, $\left( {{v_1},\;{v_4}} \right)$ , are equal in magnitude but opposite in direction. With these assumptions applied, Eq. (2) and the derivative of Eq. (4) is substituted into Eq. (5) to form the kinematics model below:

(6) \begin{align}\left[ \begin{array}{l} {\dot x} \\[2pt] {\dot y} \\[2pt] {\dot \theta } \\ \end{array} \right] = v*\left[ \begin{array}{l} {\cos (\theta )} \\[2pt] {\sin (\theta )} \\[2pt] \sigma \\ \end{array} \right],\,where\,\,\sigma = {{2\tan (\delta )} \over l}\end{align}

As shown in Fig. 7, the scaled 8WD8WS vehicle is designed to follow Ackerman’s steering geometry [Reference Adamu, Afolayan, Umaru and Garba39] to reduce tire degradation. Since the vehicle is in all-wheel steer mode, the instantaneous center of curvature is denoted by, $P$ , which intersects the CG of the vehicle. From there, eight steering angles relations shown as Eqs. (7a)–(7d) representing the kinematic constraints are calculated as where ${l_i}$ and $t$ denotes the length of each axle to the CG and the track width of the vehicle, respectively.

(7a) \begin{align}\begin{array}{*{20}{c}}{{\delta _{L1}} = {{\tan }^{ - 1}}\left( {\frac{{{l_1}}}{{R - B/2}}} \right),\;{\delta _{R1}} = {{\tan }^{ - 1}}\left( {\frac{{{l_1}}}{{R + B/2}}} \right)}\end{array}\end{align}

(7b) \begin{align}\begin{array}{*{20}{c}}{{\delta _{L2}} = {{\tan }^{ - 1}}\left( {\frac{{{l_2}}}{{R - B/2}}} \right),\;{\delta _{R2}} = {{\tan }^{ - 1}}\left( {\frac{{{l_2}}}{{R + B/2}}} \right)}\end{array}\end{align}

(7c) \begin{align}\begin{array}{*{20}{c}}{{\delta _{L3}} = {{\tan }^{ - 1}}\left( {\frac{{{l_3}}}{{R - B/2}}} \right),\;{\delta _{R3}} = {{\tan }^{ - 1}}\left( {\frac{{{l_3}}}{{R + B/2}}} \right)}\end{array}\end{align}

(7d) \begin{align}\begin{array}{*{20}{c}}{{\delta _{L4}} = {{\tan }^{ - 1}}\left( {\frac{{{l_4}}}{{R - B/2}}} \right),{\delta _{R4}} = {{\tan }^{ - 1}}\left( {\frac{{{l_4}}}{{R + B/2}}} \right)}\end{array}\end{align}

3.4. Vehicle Specifications

With the mentioned subsystems, Table I summarizes all basic dimensions and performance specifications of the SMWV.

Table I.

8WD8WS SMWV specification.

5.1. PID Differential Driving Controller

Starting with the driving controller, an incremental encoder is attached to the output shaft of each DC motor for closed-loop control, as mentioned previously. A PID controller is implemented to ensure the error between the desired and actual vehicle velocity remains as minimal as possible during operation. With the encoder feedback, the controller calculates a velocity for the CG of the vehicle, $v$ , which is used by the software differential to generate inner and outer wheel speeds in the presence of nonzero yaw commands. The goal is to improve steering maneuverability and decrease tire scrubbing. Equation (8) calculates the differential speed based on linear and angular velocity as well as the track width, $B$ , of the vehicle. Fig. 9 shows the PID control block diagram for the motors.

(8a) \begin{align}\begin{array}{*{20}{c}}{Righ{t_{velocity}} = v - \left( {\dot \theta *B/2} \right)}\end{array}\end{align}

(8b) \begin{align}\begin{array}{*{20}{c}}{Lef{t_{velocity}} = v + \left( {\dot \theta *B/2} \right)}\end{array}\end{align}

Figure 9.

PID differential driving controller block diagram.

5.2. PID Steering Controller

Once the desired steering angles are calculated based on Ackerman’s geometry from Section III, an actuator stroke position control algorithm is implemented to ensure satisfactory output. Since the vehicle features independent linear actuators for steering, built-in potentiometers are used for stroke position estimation. The maximum stroke of each linear actuator and achievable steering angles are 50 mm and 35 degrees, respectively. Figure 10 illustrates the relationship between steering angle and actuator stroke based on experimental data.

Figure 10.

Steering angle versus linear actuator stroke.

Equation (9) shows 2 third-order polynomials that model this relatively linear relationship except at full actuator extension. A control law was subsequently derived based on the PID control scheme and this model. Figure 11 shows the PID control block diagram for the linear actuators.

(9a) \begin{align}\begin{array}{*{20}{c}}{Strok{{\rm{e}}_{{\rm{left}}}} = \left( {5 \times {{10}^{ - 5}}} \right)\delta _{\rm{L}}^3 + 0.0014\delta _{\rm{L}}^2 - 0.7589{\delta _{\rm{L}}} + 24.974}\end{array}\end{align}

(9b) \begin{align}\begin{array}{*{20}{c}}{Strok{{\rm{e}}_{{\rm{right}}}} = - \left( {5 \times {{10}^{ - 5}}} \right)\delta _{\rm{R}}^3 + 0.0014\delta _{\rm{R}}^2 + 0.7586{\delta _{\rm{R}}} + 24.974}\end{array}\end{align}

Figure 11.

PID steering controller block diagram.

5.3. Incremental Localization Algorithm

To localize the SMWV, a dead reckoning algorithm that utilizes both wheel encoders and an IMU is implemented. In the proposed strategy, the wheel encoders and IMU are responsible for linear and angular displacement, respectively. The wheel encoders show acceptable performance in short-range navigation; however, IMU drift issues are hard to ignore. From the experiment, it was determined that the yaw drift is extra prominent during long-distance navigation and when the vehicle is in an immobile state. For the former issue, a linear drift was deduced during physical trials; therefore, a compensator was integrated as a remedy. To alleviate the latter issue, an algorithm that stops updating the orientation of the vehicle when it is stationary and resumes when the vehicle becomes mobile is integrated. The results show promising capabilities for the application of this work.

5.4. Path Planning

After low-level vehicle control and localization are established, two open-source algorithms are implemented to achieve high-level global and local path planning [Reference Qiu, Fan, Meng, Zhang, Cong, Li, Wang and Zhao13]. Starting with global path planning, the classic Dijkstra’s algorithm is deployed to solve the problem of generating a path between the initial and goal destination. This algorithm works by evaluating a grid cell where values are assigned to every node to represent the cost of arriving. With this information, an iterative approach is utilized to find the shortest path. To consider real-time sensor data and mobile robot dynamics, a local path planner known as the “Timed Elastic Band” is implemented. The primary goal of this algorithm is to transform a series of waypoints into a trajectory that considers time intervals. It achieves this by modifying the global planner’s output based on a multiobjective optimization framework. The objective function is the weighted summation of components, ${f_k}$ , which considers topics such as the nonholonomic constraint, fastest path, distance to waypoints, and obstacles. This is illustrated within Eq. (10) where $B$ is defined as a sequence of robot pose and time intervals, $\left( {Q,\;\tau } \right)$ :

(10) \begin{align}\begin{array}{*{20}{c}}{f\left( B \right) = \;\mathop \sum \nolimits_k {\gamma _k}{f_k}\left( B \right)\;}\end{array}\end{align}

The previously mentioned component topics are represented in two ways. For the first way, objective functions that consider the nonholonomic constraint, as well as the fastest path, are shown in Eqs. (11) and (12). In the following equation, ${d_{i,\;i + 1}}$ denotes the direction vector between two consecutive waypoints:

(11) \begin{align}\begin{array}{*{20}{c}}{{f_k}\left( {{x_i},\;{x_{i + 1}}} \right) = \;\left\|\left[ {\left( {\begin{array}{*{20}{c}}{{\rm{cos}}{\beta _i}}\\{{\rm{sin}}{\beta _i}}\\0\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{{\rm{cos}}{\beta _{i + 1}}}\\{{\rm{sin}}{\beta _{i + 1}}}\\0\end{array}} \right)} \right] \times {d_{i,\;i + 1}}\right\|^2}\end{array}\end{align}

(12) \begin{align}\begin{array}{*{20}{c}}{{f_k} = {{\left( {\mathop \sum \nolimits_{i = 1}^n \Delta{T_i}} \right)}^2}}\end{array}\end{align}

For the second way, distance to each waypoint on the generated global path as well as nearby obstacles, are considered with the following two penalty functions. Equations (11), (12), (13), and (14) are combined with Eq. (9) to form the complete multiobjective optimization framework:

(13) \begin{align}\begin{array}{*{20}{c}}{{f_{{\rm{path}}}} = e\left( {{d_{{\rm{min}},j}},\;{r_{p{\rm{max}}}},\; \in ,\;S,\;n} \right)\;}\end{array}\end{align}

(14) \begin{align}\begin{array}{*{20}{c}}{{f_{{\rm{obstacle}}}} = e\left( { - {d_{{\rm{min}},\;j}},\; - {r_{{\rm{omin}}}},\; \in ,\;S,\;n} \right)}\end{array}\end{align}

7.1. System Modeling

The forward kinematics of a two-joint manipulator is used to model the mobile robot system in this case. By establishing the relationship between different coordinate frames within the manipulator, a Jacobian matrix is derived to relate the velocity of the end effector with respect to the world, ${}_{eff}^{\phantom{ef}0}V$ , to the individual joint velocities ${\left[\begin{array}{l} {{{\dot \theta }_1}}\ {{{\dot d}_2}} \\ \end{array} \right]^T}$ . This is shown in Eq. (14) where the joint velocities, ${\left(\begin{array}{l} {{{\dot \theta }_1}}\ {{{\dot d}_2}} \\ \end{array} \right)}$ signify the robot’s angular and linear velocities, ${{{\dot \theta }_1}}$ respectively. More specifically, represents the SMWV’s angular velocity during DS and steering velocity during SS as discussed in later sections. This model restricts the movement of the robot to the x–y plane of the world frame with lateral slip neglected.

(14) \begin{align}_{eff}^{\phantom{ef}0}V = \left[ \begin{array}{l} {{v_x}} \\[3pt] {{v_y}} \\[3pt] {{v_z}} \\[3pt] {{w_x}} \\[3pt] {{w_y}} \\[3pt] {{w_z}} \\ \end{array} \right] = \left[ \begin{array}{ll} {{d_2}{C_1}} & {{S_1}} \\[3pt] {{d_2}{S_1}} & { - {C_1}} \\[3pt] 0 & 0 \\[3pt] 0 & 0 \\[3pt] 0 & 0 \\[3pt] 1 & 0 \\ \end{array} \right]\left[ \begin{array}{l} {{{\dot \theta }_1}} \\[3pt] {{{\dot d}_2}} \\[3pt] \end{array} \right] = J\left[ \begin{array}{l} {{{\dot \theta }_1}} \\[3pt] {{{\dot d}2}_1} \\ \end{array} \right]\end{align}

7.2. Stage One: Orientation Control

With the kinematics model complete, the next step is to derive the control law for stage one. To accomplish this, Lyapunov’s control scheme is applied as shown in Eq. (15) where $k$ represents the proportional gain. The error term for orientation, $e{\left( t \right)_w}$ , is the difference between the current and desired image feature sets as extracted by a pose estimation algorithm. The proposed M-PBVS algorithm formulates the control law with respect to the desired camera frame; therefore, the orientation error term is equivalent to the orientation vector, ${}_c^c\phi $ , since the desired orientation is zero. In this work, the pose estimation algorithm implemented is developed where the output is the position and orientation estimation of the visual landmark.

(15) \begin{align}e{(\dot t)_w} = - k\,e{(t)_w} = - k{{(^c}_c^d\phi - 0)_w}\end{align}

Moving forward, the derivation of the orientation control law begins with representing the angular velocity of the camera with respect to the desired camera frame, ${}_{\phantom{d}c}^{{c_d}}w$ , as the angular velocity of the camera with respect to itself, ${}_c^cw$ with the help of a rotation matrix, ${}_{\phantom{d}c}^{{c_d}}R$ .

(16) \begin{align}\begin{array}{*{20}{c}}{{}_c^cw = {{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}{}_{\phantom{d}c}^{{c_d}}w}\end{array}\end{align}

Next, a transformation matrix, $T\left( \phi \right)$ , is applied to convert the orientation expressed in Euler’s angles, ${}_{\phantom{d}c}^{{c_{\dot d}}}\phi $ , into angular velocities of the camera relative to the desired camera frame, ${}_{\phantom{d}c}^{{c_d}}w$ .

(17) \begin{align}_c^{{c_d}}w = T{(\phi )^{{c_{\dot d}}}_{\phantom{c}c}}\phi \end{align}

Accordingly, Eq. (17) is substituted into Eq. (16) to form the following.

(18) \begin{align}_c^cw = {{(^c}_c^dR)^T}T{(\phi )^{{c_{\dot d}}}_{\phantom{c}c}}\phi \end{align}

Since $e{\left( t \right)_w}$ is equivalent to ${}_{\phantom{d}c}^{{c_d}}\phi $ ; as a result, Eq. (18) is rewritten as below which considers the rate of the error, $e{(\dot t)_w}$ .

(19) \begin{align}_c^cw = {{(^c}_c^dR)^T}T(\phi )*e{(\dot t)_w}\end{align}

By substituting Lyapunov’s control scheme from Eq. (15) into Eq. (19), the following control law for the angular velocity of the camera frame is derived.

(20) \begin{align}\begin{array}{*{20}{c}}{{}_c^cw = - k\;{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}T\left( \phi \right)*e{{\left( t \right)}_w}}\end{array}\end{align}

Since the SMWV is only pivoting without translation at this stage; therefore, the two joint manipulator model is reduced to just the revolute joint. As a result, the angular velocity of the SMWV’s base is related to the angular velocity of the camera as shown in Eq. (21) where, ${}_c^bR$ , represent an identity rotation matrix because the camera is rigidly attached to the base.

(21) \begin{align}\begin{array}{*{20}{c}}{{}_b^bw = {}_c^bR\;{}_c^cw}\end{array}\end{align}

Next, the angular velocity of the base, ${}_b^bw$ , is related to the angular velocity of the base with respect to the world, ${}_b^0w$ by Eq. (22).

(22) \begin{align}\begin{array}{*{20}{c}}{{}_b^0w = {}_b^0R\;{}_b^bw}\end{array}\end{align}

Using Eq. (14), the angular velocity of the SMWV’s base frame is described based on the angular joint velocity, $\dot \theta $ , using a Jacobian matrix, ${J_w}$ , as seen below.

(23) \begin{align}\begin{array}{*{20}{c}}{{}_{{\rm{base}}}^{\phantom{bas}0}w = {J_w}\;\dot \theta }\end{array}\end{align}

By combining Eqs. (20), (21), (22), and (23), the complete law that controls the angular velocity of the SMWV based on the orientation error is shown below. It is important to note that a pseudo-inverse for the Jacobian matrix, $J_w^ + $ , is applied to approximate the inverse kinematics.

(24) \begin{align}\begin{array}{*{20}{c}}{\dot \theta = - k\;J_w^ + \;{}_b^0R\;{}_c^bR\;{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}\;T\left( \phi \right)\;e{{\left( t \right)}_w}}\end{array}\end{align}

Using differential kinematics, the angular velocity of the SMWV is further expressed in terms of wheel velocities as shown in Eq. (25).

(25) \begin{align}\begin{array}{*{20}{c}}{\dot \theta = \;\left[ {\begin{array}{*{20}{c}}{ - \dfrac{r}{L}} {\dfrac{r}{L}}\end{array}} \right]\;\left[ {\begin{array}{*{20}{c}}{{\omega _{{\rm{left}}}}}\\[3pt] {{\omega _{{\rm{right}}}}}\end{array}} \right]}\end{array}\end{align}

7.3. Stage Two: Position Control

Stage two of the M-PBVS algorithm begins with first determining which of the two scenarios the SMWV is currently in (direct or alternate goal). To do this, the desired position coordinates, $\left( {{x_d},\,{y_d}} \right)$ , are compared with the current, $\left( {x,\,y} \right)$ as shown in Eq. (26). The approach angle is then measured against the maximum steering angle to determine which is larger in value.

(26) \begin{align}{{\delta _{{\rm{approach}}}} = \;\dfrac{{{y_d} - y}}{{{x_d} - x}}}\end{align}

If the approach angle is less than the maximum steering angle then the SMWV is in the direct goal scenario where the steering angles are set as shown below.

(27) \begin{align}{{\delta _{{\rm{steering}}}} = {{\tan }^{ - 1}}{\delta _{{\rm{approach}}}}}\end{align}

If the approach angle is greater than the maximum steering angle, then the SMWV requires an alternate goal which is calculated by finding the intersection between lines extended from the current and desired positions at the maximum steering angle as calculated by Eqs. (26) and (27). These lines are represented by grey dash lines in Fig. 12.

(28) \begin{align}\begin{array}{*{20}{c}}{\textrm{current:y} = \;\tan \left( { \pm {\delta _{{\rm{max}}}}} \right)\;x + b}\end{array}\end{align}

(29) \begin{align}\begin{array}{*{20}{c}}{\textrm{desired:}{y_d} = \;\tan \left( { \pm {\delta _{{\rm{max}}}}} \right)\;{x_d} + {b_d}}\end{array}\end{align}

By doing so, the alternate goal is guaranteed to be within the achievable steering angle. Once the SMWV has arrived at the alternate goal, the new approach angle towards the desired goal should be equivalent to the maximum steering angle; thereby making it a direct goal. It is important to note that there are two alternate goal solutions at each initial position because of paths calculated based on positive and negative ${\delta _{{\rm{max}}}}$ . To solve this issue, the alternate goal with the shortest distance from its current position is always selected. For example, alternate goal scenarios in Quadrant 1 and 2 would always result in the SMWV generating a forward velocity as shown by Position D in Fig. 12. On the other hand, the opposite is true for alternate goal scenarios in Quadrants 3 and 4 as illustrated by Positions A and E where the shortest distance requires the SMWV to reverse. Once the direct or alternate goal positions are either detected or calculated, Lypapunov’s proportional control scheme is applied similarly to stage one. However, the only difference is that the error term in the second stage consists of two vectors which are the translational vector, ${}_{\phantom{d}c}^{{c_d}}s$ , and the orientation vector, ${}_{\phantom{d}c}^{{c_d}}\phi $ . Starting with the latter, the derivation of the orientation control for stage two is the same as stage one; however, steering velocity, $\dot \varphi $ , is used for SS instead of the angular velocity from Eq. (24). Also, the revolute joint’s angle is constrained by the maximum steering angle. The following illustrates the steering control law.

(30) \begin{align}\begin{array}{*{20}{c}}{\dot \varphi = - k\;J_w^ + \;{}_b^0R\;{}_c^bR\;{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}\;T\left( \phi \right)\;e{{\left( t \right)}_w}}\end{array}\end{align}

where $ - {\delta _{{\rm{max}}}} \lt {\theta _1} \lt {\delta _{{\rm{max}}}}$ . On the other hand, Lyapunov’s control scheme is applied to the translational vector similarly to Eq. (15) for orientation as shown below.

(31) \begin{align}e{(\dot t)_t} = - k{{(^c}_c^ds - 0)_s}\end{align}

Next, the translational vector is represented in terms of translational camera velocity, ${}_c^cv$ , as shown below.

(32) \begin{align}{}_c^cv = {{({}^c}_c^dR)^T}^{\phantom{d}c}_c{}^{\dot d}s\end{align}

From here, the rate of change of the translational vector, $^c_c{}^{\dot d}s$ , is equivalent to the rate of the error as suggested by Eq. (31). As a result, the translational control law is written as follows.

(33) \begin{align}\begin{array}{*{20}{c}}{{}_c^cv = \; - k{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}e{{\left( t \right)}_s}}\end{array}\end{align}

The above control law is capable of controlling the translational velocity of the camera; however, it is not complete in the sense that it does not consider SMWV’s base frame. Therefore, rotation matrices between the world and camera frame are substituted into Eq. (31) to formulate the translational control law below.

(34) \begin{align}\begin{array}{*{20}{c}}{v = \; - k\;J_t^ + {}_b^0R\;{}_c^bR\;{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}\;e{{\left( t \right)}_s}}\end{array}\end{align}

Lastly, the full control law for stage two of the proposed M-PBVS is completed by combining the steering control from Eq. (28) and the translational control as seen above to form the following where $s(d)$ is a skew matrix that describes the camera’s position with respect to the base frame.

(35) \begin{align}\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}v\\{\dot \varphi }\end{array}} \right] = \; - k\;{J^ + }\;E\;L\;e\left( t \right)}\end{array}\end{align}

(36) \begin{align}\begin{array}{*{20}{c}}{E = \;\left[ {\begin{array}{c@{\quad}c}{{}_b^0R} & {{0_{3x3}}}\\[3pt] {{0_{3x3}}} & {{}_b^0R}\end{array}} \right]\left[ {\begin{array}{c@{\quad}c}{{}_c^bR} & {s\left( d \right){}_c^bR}\\[3pt] {{0_{3x3}}} & {{}_c^bR}\end{array}} \right]}\end{array}\end{align}

(37) \begin{align}\begin{array}{*{20}{c}}{L = \left[ {\begin{array}{c@{\quad}c}{{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}} & 0\\[3pt] 0 & {{{\left( {{}_{\phantom{d}c}^{{c_d}}R} \right)}^T}{\rm{*}}T\left( \phi \right)}\end{array}} \right]}\end{array}\end{align}

7.4. Mobile Robot Parking Algorithm

As mentioned previously, a common limitation in recent literature regarding autonomous parking is the lack of obstacle avoidance and the use of outdated sensors. Additionally, specific coordinates of the charging station are often required. These two requirements cause fundamental problems since obstacle and parking station locations can frequently change depending on the environment. To combat these constraints, the proposed algorithm works in two stages. First, the path planners are utilized to generate a collision-free path towards the parking area. Once arrived, the SMWV utilizes the proposed M-PBVS algorithm to correct its pose and reach the desired location precisely. This method allows the SMWV to plan its path online while specific parking station coordinates are not necessary. The pseudocode of the proposed mobile robot parking approach is presented in Table IV.

Table IV.

Mobile robot parking pseudocode

8.1. Experiment Set-Up

As previously mentioned, the vehicle is instrumented with a 360-degree laser scanner and IMU for obstacle detection and localization, respectively. Figure 13 shows the sensor placement relative to the base frame of the vehicle. The laptop is placed on top of the chassis at the center of the vehicle. The two experiments are conducted in an indoor environment with flat smooth surfaces and opaque, rigid obstacles. Due to the placement of the laser scanner at the top of the chassis, a minimum obstacle height of 0.6 m is necessary for obstacle detection during navigation.

Figure 13.

Experimental setup.

8.2. Slalom Test

Starting with the Slalom Test, the vehicle begins at the origin with a goal at approximately $\left( {10,\; - 0.8} \right)$ in the map frame. Two barrier obstacles are placed along the way with dimensions of approximately 1.0 × 0.6 m. Figure 14(a) successfully illustrates the robot’s trajectory as generated by both Dijkstra and timed elastic band (TEB), around the obstacles towards its goal. The whole navigation took approximately 30 s with the maximum linear and angular velocities constrained to 0.3 m/s and 0.25 rad/s, respectively. In Fig. 14(b), the desired velocity remained at the maximum velocity throughout the course until the last 5 s where the vehicle slowed down to adjust its pose. Since the desired velocity represents the output of the TEB local planner, the actual velocity data exhibits significantly more noise as it is obtained from achieved wheel speeds. Regardless of the noise, it is apparent that the vehicle was able to follow the path planner commands as the overall trends of both data resemble each other. In the same graph, the inner and outer wheel velocities are also displayed. In this case, the left and right wheel velocities zig-zag one another as the angular velocities from Fig. 14(c) are taken into account. In this figure, the vehicle attempts to clear the first and second obstacle between 0–18 s and 18–28 s, respectively. By the convention assigned in the experimental setup section, a negative angular velocity implies a right steer command. With this in mind, it is easy to see that the differential speed from Fig. 14(b) matches accordingly as outer wheels always exhibit a higher velocity than inner due to turning radius difference. In Fig. 14(d), the steering angles of the front two axles are illustrated. It is important to note that the rear steering angles exhibit the same magnitude but opposite in direction (for spacing and clarity, only the front two-axle steering angles are displayed). From this figure, the maximum steering angle reached is 35 degrees. When looking closer at Fig. 14(d) around 25 s, the vehicle is attempting to steer right after clearing the second obstacle.

Figure 14.

(a) robot trajectory, (b) linear velocity, (c) front two-axle steering angles, (d) angular velocities.

During this, the first axle right wheel exhibits the highest turning angle, followed by the left wheel of the same axle and then the right and left wheels of the second axle, accordingly. This relationship represents the Ackerman geometry that was discussed in the previous sections. Figure 15 shows consecutive images of the slalom experiment with the top row displaying the physical SMWV and the bottom row displaying the accompanying laser data visualization. This test shows the overall successful navigation ability of the vehicle in an obstacle-ridden environment using its steering capabilities and arriving at the destination within a minimal tolerance with regards to both position and orientation.

Figure 15.

Physical experiment (top) and RVIZ (bottom).

8.3. Parking experimentation

To validate the proposed two-stage Parking algorithm, first, the navigation ability of the SMWV is tested. From there, the proposed M-PBVS algorithm is employed to alleviate any pose inaccuracies. Lastly, the M-PBVS algorithm is compared with traditional path planning to study the improvements made.

Figure 16.

Autonomous navigation experiment setup.

Figure 17.

Physical experiment (top) and RVIZ (bottom).

8.3.1. Autonomous Navigation Test

This test intends to evaluate stage one of the proposed algorithms which is the SMWV’s ability to navigate towards the parking area. To do so, three initial regions that are located throughout an indoor hallway as seen in Fig. 16 are chosen. The map of the test environment was acquired prior to the experimentation using standard ROS packages, namely the Open Slam implementation known as Gmapping [Reference Koubaa40]. During the experiment, the map was loaded and the robot’s initial position was manually defined prior to autonomous travel. Within each region, three different positions and orientations are randomly selected as starting poses. From there, all poses are sent to the parking area via Dijkstra’s and TEB algorithm to account for any obstacles along the way [Reference Magyar, Tsiogkas, Deray, Pfeiffer and Lane41,Reference Mao and Ma42]. To ensure a proper reset of error, sensor power is cycled between tests. When looking closer at one of the trials, Fig. 16 illustrates the path planner’s ability to utilize the onboard sensors to successfully navigate around obstacles towards its goal. Figure 17 shows consecutive images of this experiment with the top row displaying the physical SMWV and the bottom row displaying the accompanying laser data visualization. Since the accuracy of the final achieved pose is of great importance in parking applications, the test is repeated nine times. Taking a look at the overall trend, Table V tabulates the different starting poses as well as the final achieved poses with their respective average position and orientation error. From these results, it is evident that Region 1 exhibits the highest amount of drift, leading to the largest deviation of 2.06 m and 0.26 rad between achieved and desired. Region 3 achieves a more accurate result with a deviation of 0.98 m and 0.19 rad. The final positions are plotted in Fig. 18, which shows the tolerance window that is calculated by taking the average error of all final positions. The result is a circle with a 1.5-m radius. Based on the findings of this experiment, it was concluded that the SMWV can navigate autonomously; however, with an average position and orientation error of 1.54 m and 0.22 rad, respectively.

Table V.

Navigation test results

Figure 18.

Drift results.

8.3.2. M-PBVS Test

With the findings of the previous experiment, the following evaluates the M-PBVS algorithm’s performance in terms of close quarters pose correction. Starting with the experimental setup, all tests conducted in this section are performed in an indoor lab environment with smooth surfaces. The desired position is chosen in front of a visual landmark at (0, 0) with a heading angle of zero as shown in Fig. 19. Note here that the heading angle, $\vartheta $ , is measured relative to the longitudinal axis of the SMWV where a counter-clockwise rotation is deemed positive. From the desired pose, four quadrants are identified based on a cartesian coordinate system. When considering the results of the previous experiment as well as the setup presented here, two sets of experiments are conducted. The first test known as the M-PBVS Test begins with the SMWV at a position and orientation error that fits within the previously acquired result. From there, it is the M-PBVS’s responsibility to determine whether it is in a direct or alternate goal scenario and then subsequently dock the SMWV as precisely as it can. The second test is called the Comparison Test which studies the performance differences between the proposed M-PBVS algorithm and the traditional path planner. Experimental data such as trajectory, linear/angular velocity as well as position and orientation errors are presented.

Figure 19.

Experimental setup.

Figure 20.

Direct goal trajectory.

Starting with the M-PBVS test, position and orientation errors are introduced by placing the SMWV at position (0.15, 0.9) with a –0.2 rad heading angle in Quadrant 1. As mentioned before, the desired position is the origin of the cartesian plane with a 0 rad heading angle. In Fig. 20, the initial and final pose of the SMWV is illustrated on top of the trajectory that is generated by the proposed M-PBVS algorithm. Beginning with the first stage of the M-PBVS algorithm where an angular velocity is generated to reduce the heading error with the DS configuration. This velocity is evident during the first 4 s of the test where it reached a maximum of 0.15 rad/s, pivoting the SMWV clockwise about its center as seen in Fig. 21. Because of this, the orientation error is reduced to approximately zero as shown in Fig. 22 while both the linear velocity and position error remain unchanged. Once the orientation is corrected, the SMWV enters the second stage of M-PBVS which first determines whether it is in a direct goal or alternate goal scenario. Based on the calculation from Eq. (26), the initial position qualifies as a direct goal scenario as the initial ${\delta _{{\rm{approach}}}}$ is approximately 10 degrees which is less than the $\delta {\rm{max}}$ . Consequently, the generated steering angles are shown by Fig. 23 which remains between 8 and 10 degrees in a SS configuration until it arrives at the intended goal. In parallel with the steering control, the linear velocity reached a maximum of 0.2 m/s before slowly reducing to zero over the next 20 s. Also, the angular velocity is zero while the linear velocity is negative because the desired pose is set to behind the initial as shown in Fig. 22 during the second stage. Snapshots of the physical experiment along with the camera’s field of view are shown in Fig. 24. The result from this test successfully illustrates the M-PBVS’s ability to correct SMWV’s initial pose to centimeter accuracy as the final achieved position is (0.01, –0.04) which yields an error percentage of 5.6%.

Table VI.

Comparison results.

Figure 21.

Direct goal velocity.

Figure 22.

Direct goal pose error.

Figure 23.

Direct goal steering angle.

Figure 24.

Direct goal test: physical experiment (top) and camera view (bottom).

Next is the Comparison Test which is intended to gain a better insight into the performance difference between TEB and M-PBVS. This time, the initial pose is located at (0.64, –0.91) within Quadrant 2 with an initial orientation of –0.28 rad. This pose is chosen such that the M-PBVS algorithm would enter an alternate goal scenario to avoid showing similar results as the previous test. The trajectory of both algorithms is plotted in Fig. 25 where the TEB planner changes the direction of the SMWV a total of four times while the M-PBVS algorithm generates a path that only requires a change in direction once. The result of this is a much higher traveled distance of 8.91 m for the TEB when compared to the 1.85 m by the M-PBVS algorithm. Furthermore, the M-PBVS’s efficiency is also prevalent when looking at Fig. 26 where the position error reduced to zero in 11 s compared to 19 s from the TEB planner. Also, the final achieved position of the TEB and M-PBVS algorithm is (0.23, 0.17) and (0.02, –0.05), respectively, which shows that the M-PBVS algorithm is much more accurate at position correction. In addition, the orientation error from Fig. 27 further validates the ability of the M-PBVS algorithm as it reduces the orientation error within 3 s with an error of 0.71%. The TEB planner utilizes multiple turns to correct its orientation yet its final result yields less accuracy than the M-PBVS algorithm with an error of 60.71%. From both comparison tests in this section, it is evident that the simplicity of the M-PBVS algorithm achieves higher accuracy and efficiency in both position and orientation correction when compared with the TEB planner. The following table (Table VI) tabulates the performance metrics between the two algorithms during this test.

Figure 25.

Alternate goal trajectory comparison.

Figure 26.

Alternate goal position error comparison.

Figure 27.

Alternate goal orientation error comparison.