2.1. Robot structure

As it illustrated in Fig. 1, the WLR analyzed in this study features 8 degrees of freedom, with 2 degrees of freedom allocated to each joint (wheel-leg). Constructed from aluminum sheets and plexiglass, the robot’s body incorporates an aluminum sheet in the lower section, which serves to lower the center of gravity and house electronic components, while plexiglass sheets are utilized as body covers. The body dimensions measure 7 × 35 × 32 cms, with the legs measuring 13.5 × 5 × 5 cms. The robot’s weight, excluding the power source, amounts to 3 kgs. The wheels, with a diameter of 6.8 cms, are specifically designed with a tread pattern to enhance rotation on sloping and slippery surfaces, thereby augmenting the wheels′ friction with the terrain. Mechatronic components include 8 Dynamixel motors, each with a power rating of 150 W; this comprises four XM540-W150-R motors for hip joints and four XL430-W250-T motors for the wheels. These motors are capable of transmitting feedback on position, speed, and torque to the PC. The internal motor gearboxes directly link the wheels and legs, while the electronic elements are contained in the main body of the robot. An accelerometer and gyroscope, the MPU-6050, serve as a 6-axis IMU sensor to determine the posture and position of the robot’s center of mass. The PC acts as the main controller for the robot, receiving all analog and digital inputs and outputs, allowing commands to be issued to the actuators of the wheels and legs. Table I shows the actual parameters of WLR.

Table I.

Actual parameters of WLR.

Figure 1.

Schematic of the robot components and its electronic circuit.

2.2. Vision method

The effect of slippage on wheeled robots during accelerated movements can result in significant errors between the expected and actual outcomes. Fig. 2 illustrates the application of pose estimation method along with concentric circles to assess the displacement of robot movement and body slippage in a controlled laboratory setting. To address the challenges posed by noise and varying lighting conditions, this method was employed to track the position of the robot body in captured images. By identifying and averaging the coordinates of the centers of the circles, a virtual coordinate system was established at the midpoint of the circles on the robot body, enabling the calculation of displacement for that specific point. By comparing the robot’s movement trajectory with the intended path, we are able to quantify the extent of body slippage. Deviation from the path and failure to reach the final destination of the robot is caused by two factors, slipping and skidding. The direction vectors of these two factors are opposite to each other, and separating them from each other is difficult. Therefore, these two factors have been measured together as the slip of the robot’s wheels and have been incorporated into the robot’s equations.

Figure 2.

Schematic of the slip estimation process.

3.1. Kinematics of the WLR

By writing the position vectors of the wheels relative to the center of mass of the robot and taking derivatives of them, we determine the relative speeds of the wheels relative to its center of mass and obtain the speeds of the wheels and legs of the robot (point $O$ ). In Eq. (1a), $\boldsymbol{v}$ is the speed of the center of the wheels and legs of the robot (point $O$ ) [Reference Nikseresht, Korayem and Toorani30].

(1) \begin{align}\boldsymbol{v}=\boldsymbol{v}_{\boldsymbol{G}}+\dot{\theta }\hat{\boldsymbol{k}}\times {\boldsymbol{r}}_{{\boldsymbol{O}}/{\boldsymbol{G}}}+\boldsymbol{v}_{{\boldsymbol{O}}/{\boldsymbol{G}}}\end{align}

where $\boldsymbol{v}_{\boldsymbol{G}}$ represents the center of mass velocity of the platform, ${\boldsymbol{r}}_{{\boldsymbol{O}}/{\boldsymbol{G}}}$ denotes the vector connecting the wheel center to the center of mass, and $\boldsymbol{v}_{{\boldsymbol{O}}/{\boldsymbol{G}}}$ signifies the relative velocity of the wheel with respect to the platform. We decompose the robot’s body velocity along the coordinate axes as follows:

(2) \begin{align}\left(\boldsymbol{v}_{\boldsymbol{G}}\right)_{x}=\dot{x},\left(\boldsymbol{v}_{\boldsymbol{G}}\right)_{y}=\dot{y},\left(\boldsymbol{v}_{\boldsymbol{G}}\right)_{z}=\dot{z}\end{align}

By writing Eq. (1) for each of the wheels and incorporating the linear velocities of the wheels and their slip components on the opposite side, the kinematic equations for the robot, assuming longitudinal, lateral and vertical slips, take the form of Eq. (3– 6):

Front right wheel velocity:

(3) \begin{align}\boldsymbol{v}_{\boldsymbol{rf}}= \left(\dot{x}-\dot{x}_{srf}-a\dot{\theta }_{rf}\right)\hat{\boldsymbol{i}} + \left(\dot{y}-\dot{y}_{srf}\right)\hat{\boldsymbol{j}} + \left(\dot{z}-\dot{z}_{srf} \right)\hat{\boldsymbol{k}}+\dot{\theta }\hat{\boldsymbol{k}}\times {\boldsymbol{r}}_{\boldsymbol{rf}}+\dot{{\boldsymbol{r}}}_{\boldsymbol{rf}}\end{align}

Front left wheel velocity:

(4) \begin{align}\boldsymbol{v}_{\boldsymbol{lf}}= \left(\dot{x}-\dot{x}_{slf}-a\dot{\theta }_{lf}\right)\hat{\boldsymbol{i}} + \left(\dot{y}-\dot{y}_{slf}\right)\hat{\boldsymbol{j}} + \left(\dot{z}-\dot{z}_{slf}\right)\hat{\boldsymbol{k}}+\dot{\theta }\hat{\boldsymbol{k}}\times {\boldsymbol{r}}_{\boldsymbol{lf}}+\dot{{\boldsymbol{r}}}_{\boldsymbol{lf}}\end{align}

Rear right wheel velocity:

(5) \begin{align}\boldsymbol{v}_{\boldsymbol{rr}}= \left(\dot{x}-\dot{x}_{srr}-a\dot{\theta }_{rr}\right)\hat{\boldsymbol{i}} + \left(\dot{y}-\dot{y}_{srr}\right)\hat{\boldsymbol{j}} + \left(\dot{z}-\dot{z}_{srr}\right)\hat{\boldsymbol{k}}+\dot{\theta }\hat{\boldsymbol{k}}\times {\boldsymbol{r}}_{\boldsymbol{rr}}+\dot{{\boldsymbol{r}}}_{\boldsymbol{rr}}\end{align}

Rear left wheel velocity:

(6) \begin{align}\boldsymbol{v}_{\boldsymbol{lr}} = \left(\dot{x}-\dot{x}_{slr}-a\dot{\theta }_{lr}\right)\hat{\boldsymbol{i}} + \left(\dot{y}-\dot{y}_{slr} \right)\hat{\boldsymbol{j}} + \left(\dot{z}-\dot{z}_{slr}\right)\hat{\boldsymbol{k}}+\dot{\theta }\hat{\boldsymbol{k}}\times {\boldsymbol{r}}_{\boldsymbol{lr}}+\dot{{\boldsymbol{r}}}_{\boldsymbol{lr}}\end{align}

In the following, we proceed by differentiating the above relations and determining the accelerations to formulate the dynamic equations.

3.2. Dynamics of the WLR

The Gibbs–Appell method has been used to obtain the dynamic equations. The pseudo-coordinates of the problem are determined and then the system accelerations are extracted in terms of pseudo-coordinates. Next, the Gibbs functions for different parts of the robot are determined and by putting them into the Gibbs equation and taking derivatives with respect to each of the pseudo accelerations, the dynamic equations of the robot are extracted. The pseudo-coordinates are $\alpha$ , $\beta$ , $\theta _{rf}$ , $\theta _{lf}$ , $\theta _{rr}$ , $\theta _{lr}$ , $x_{slf}$ , $x_{srf}$ , $x_{slr}$ , $x_{srr}$ , $y_{l}$ , $y_{r}$ with derivatives from the Gibbs–Appell equations in terms of pseudo accelerations, which are second-order derivatives of the pseudo-coordinates, the governing dynamic equations of the system are obtained. The Gibbs functions are determined separately for the three main parts of the robot.

Gibbs function for wheels:

(7) \begin{align}S_{1}=\frac{I_{o}}{2} \left(\ddot{\theta }_{rf}^{2}+\ddot{\theta }_{lf}^{2}+\ddot{\theta }_{rr}^{2}+\ddot{\theta }_{lr}^{2}\right)\end{align}

Gibbs function for legs:

(8) \begin{align}S_{2}=I_{ly}\left(\ddot{\alpha }^{2}+\ddot{\beta }^{2}\right)+I_{lx}\ddot{\theta }^{2}\left(cos^{2}\alpha +cos^{2}\beta \right)+\frac{m_{l}}{2} \left(a_{rf}^{2}+a_{lf}^{2}+a_{rr}^{2}+a_{lr}^{2}\right)\end{align}

Gibbs function for the body [Reference Toorani, Korayem and Davaie Markazi7]:

(9) \begin{align}S_{3}=\frac{1}{2} \left( I_{Gz}\ddot{\theta }^{2} + I_{Gy}\ddot{\Psi}^{2} + M_{G} {{\boldsymbol{a}}_{\boldsymbol{G}}}^{\mathbf{2}}\right)\end{align}

The final Gibbs function is obtained by combining Eq. (7– 9):

(10) \begin{align} S =&\, \frac{I_{o}}{2}\left(\ddot{\theta }_{rf}^{2}+\ddot{\theta }_{lf}^{2}+\ddot{\theta }_{rr}^{2}+\ddot{\theta }_{lr}^{2}\right)+I_{ly}\left(\ddot{\alpha }^{2}+\ddot{\beta }^{2}\right)+I_{lx}\ddot{\theta }^{2}\left(cos^{2}\alpha +cos^{2}\beta \right)+\frac{m_{l}}{2}\left(a_{rf}^{2}+a_{lf}^{2}+a_{rr}^{2}+a_{lr}^{2}\right)\nonumber\\&+\frac{1}{2}\left(I_{Gz}\ddot{\theta }^{2}+I_{Gy}\ddot{\Psi }^{2}+M_{G}\boldsymbol{a}_{\boldsymbol{G}}^{\mathbf{2}}\right)\end{align}

In the above equations, $I_{o}$ is the wheel axis moment of inertia, $I_{ly}$ is the leg $Y$ -axis moment of inertia, $I_{lx}$ is the leg $X$ -axis moment of inertia, $m_{l}$ is the leg mass, $a_{lf}$ , $a_{rf}$ , $a_{lr}$ , $a_{rr}$ are the accelerations of each leg, $I_{Gz}$ is the body $Z$ -axis moment of inertia, $I_{Gy}$ is the body $Y$ -axis moment of inertia, $M_{G }$ is the robot body mass, and $\boldsymbol{a}_{\boldsymbol{G}}$ is the acceleration of the robot’s center of mass, which is calculated as a vector from the following equation:

(11) \begin{align}\boldsymbol{a}_{\boldsymbol{G}}=\ddot{x}\hat{\boldsymbol{i}}+\ddot{y}\hat{\boldsymbol{j}}+\ddot{z}\hat{\boldsymbol{k}}+\dot{\theta }\hat{\boldsymbol{k}}\times \dot{x}\hat{\boldsymbol{i}}\end{align}

In which, $\ddot{x}$ , $\ddot{y}$ , $\ddot{z}$ represent the accelerations of the robot’s center of mass along the coordinate axes. Subsequently, the potential energy function of the system is calculated as follows:

(12) \begin{align}V=gl(m_{l}\left(sin\left(\alpha \right) + sin\left(\beta \right)\right)+\frac{M_{G}}{2}(sin\left(\alpha +\Psi \right)+sin\left(\beta +\Psi \right)))\end{align}

where $g$ represents the acceleration due to gravity. To determine dynamic slip or traction force, due to the complexity of relationships, a simplified model of the Dahl friction is employed under quasi-static conditions, which transforms into the Coulomb friction equation [Reference Pennestrì, Rossi, Salvini and Valentini31]:

(13) \begin{align}F_{slip}=-\sigma _{0}\hat{z}\end{align}

Here, $\sigma _{0}$ denotes the surface roughness coefficient, and $\hat{z}$ represents relative displacement. In the static case, Eq. (13) becomes the following expression for Coulomb friction force [Reference Marques, Flores, Pimenta Claro and Lankarani32]:

(14) \begin{align}F_{slip}=-\mu _{d}F_{n}sgn\left(\upsilon \right)\end{align}

where $\mu _{d}$ is the coefficient of dynamic friction, $F_{n}$ is the normal force on the wheel surface, and $\upsilon$ is the linear velocity of the wheels. The friction function for the Gibbs–Appell equations is defined as:

(15) \begin{align} D =&\, \mu _{dx}(F_{nf}\left(\dot{x}_{srf}sgn\left(\dot{x}_{srf}\right)+\dot{x}_{slf}sgn\left(\dot{x}_{slf}\right)\right)+F_{nr}\left(\dot{x}_{srr}sgn\left(\dot{x}_{srr}\right)+\dot{x}_{slr}sgn\left(\dot{x}_{slr}\right)\right) \nonumber\\& + 2\mu _{dy}(F_{nf}\dot{y}_{f}sgn\left(\dot{y}_{f}\right)+F_{nr}\dot{y}_{r}sgn\left(\dot{y}_{r}\right))\end{align}

In this equation, $\mu _{dx}$ represents the longitudinal friction coefficient of the wheels, $\mu _{dy}$ represents the lateral friction coefficient of the wheels also $F_{nf}$ and $F_{nr}$ are the normal forces on the front and rear wheels, respectively. Additionally, in Eq. (15), a velocity term is multiplied by sign functions, which, by taking derivatives from them, we will obtain the form of frictional force in the Gibbs-Appell equations. The normal forces on the wheel surfaces are obtained from the following relationships.

Front wheel normal force:

(16) \begin{align}F_{nf} =&\, \frac{1}{l_{x}cos\left(\Psi \right)+l(cos\left(\alpha +\Psi \right)-cos\left(\beta +\Psi \right))}(\frac{1}{2}M_{G}g(\frac{1}{2}l_{x}cos\left(\Psi \right)-lcos\left(\beta +\Psi \right)) \nonumber\\& + m_{l}g(l_{x}cos\left(\Psi \right)+\frac{1}{2}lcos\left(\alpha +\Psi \right)-\frac{3}{2}lcos\left(\beta +\Psi \right)))\end{align}

Rear wheel normal force:

(17) \begin{align}F_{nr} = & \frac{1}{l_{x}cos\left(\Psi \right)+l(cos\left(\alpha +\Psi \right)-cos\left(\beta +\Psi \right))}(\frac{1}{2}M_{G}g(\frac{1}{2}l_{x}cos\left(\Psi \right)+lcos\left(\alpha +\Psi \right)) \nonumber\\ & + m_{l}g(l_{x}cos\left(\Psi \right)+\frac{3}{2}lcos\left(\alpha +\Psi \right)-\frac{1}{2}lcos\left(\beta +\Psi \right)))\end{align}

After formulating the Gibbs functions, potential function and friction function, these functions are incorporated into the final Gibbs equations to provide the dynamic equations for all pseudo-coordinates. The general form of the Gibbs equation, considering slip, is defined as

(18) \begin{align}\frac{\partial S}{\partial \ddot{q}}+\frac{\partial D}{\partial \dot{q}}+\frac{\partial V}{\partial q}=\tau\end{align}

where $q$ represents the pseudo-coordinates of the system, and $\tau$ denotes the generalized torques (inputs). By incorporating each input into the equation above and taking derivatives, the governing dynamic equations for the system are derived. Notably, since 12 pseudo-coordinates have been obtained as independent coordinates, a set of 12 equations serves as the dynamic model for the robot. Furthermore, the formulation of the Gibbs–Appell method, Lagrange method, and their comparative analysis has been addressed in ref [Reference Desloge33]. Next, by substituting pseudo accelerations in place of $q$ for torques and forces acting on the system, the following quantities are extracted.

Torque of front leg motors:

(19) \begin{align}\tau _{\alpha } = &\, 2I_{ly}\ddot{\alpha }+\frac{1}{2} \left( m_{l} \left({-}2\left(\ddot{x}-\ddot{x}_{srf}-a\ddot{\theta }_{rf}+\frac{1}{2}\left(\ddot{\theta }l_{y}-l_{x}\ddot{\psi } {sin} \left(\psi \right)-l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)\right)\right.\right.\right. \nonumber\\&\left.\left.\left. -l\left(\ddot{\alpha }+\ddot{\psi }\right) {sin} \left(\alpha +\psi \right)-l\left(\dot{\alpha} + \dot{\psi }\right)^{2} {cos} \left(\alpha +\psi \right)\right) l {sin} \left(\alpha +\psi \right)\right.\right. \nonumber\\& \left.\left.+ \left(\ddot{z}-\ddot{z}_{srf}+\frac{1}{2} \left(l_{x}\ddot{\psi } {cos} \left(\psi \right)-l_{x}\dot{\psi }^{2} {sin} \left(\psi \right)+l\left(\ddot{\alpha }+\ddot{\psi }\right) {cos} \left(\alpha +\psi \right)-l\left(\dot{\alpha }+\dot{\psi }\right)^{2} {sin} \left(\alpha +\psi \right)\right)\right.\right.\right. \nonumber \\& \left.\left.\left.l\,{cos} \left(\alpha +\psi \right)-2\left(\ddot{x}-\ddot{x}_{slf}-a\ddot{\theta }_{lf}+\frac{1}{2} \left({-}\ddot{\theta }l_{y}-l_{x}\ddot{\psi } {sin} \left(\psi \right)-l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)\right)-l\left(\ddot{\alpha }+\ddot{\psi }\right)\right. \right.\right.\right.\nonumber\\ & \left.\left.{sin} \left(\alpha +\psi \right)-l\left(\dot{\alpha }+\dot{\psi }\right)^{2} {cos} \left(\alpha +\psi \right)\right) l {sin} \left(\alpha +\psi \right) + \left(\ddot{z}-\ddot{z}_{slf}+\frac{1}{2} \left(l_{x}\ddot{\psi } {cos} \left(\psi \right)-l_{x}\dot{\psi }^{2}\right. \right.\right.\nonumber\\ &\left. \left.\left. {sin} \left(\psi \right)+l\left(\ddot{\alpha }+\ddot{\psi }\right) {cos} \left(\alpha +\psi \right)-l\left(\dot{\alpha }+\dot{\psi }\right)^{2} {sin} \left(\alpha +\psi \right)\right)l\,{cos} \left(\alpha +\psi \right)\right)\right)\end{align}

Torque of rear leg motors:

(20) \begin{align} \tau _{\beta } =&\, 2I_{ly}\ddot{\beta }+\frac{1}{2}\left(m_{l}\left({-}2\left(\ddot{x}-\ddot{x}_{srr}-a\ddot{\theta }_{rr}+\frac{1}{2}\left(\ddot{\theta }l_{y}+l_{x}\ddot{\psi } {sin} \left(\psi \right)+l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)\right)\right.\right.\right. \nonumber\\ & \left. \left.\left.\left.-l\left(\ddot{\beta }+\ddot{\psi }\right) {sin} \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2} {cos} \left(\beta +\psi \right)\right)l {sin} \left(\beta +\psi \right)+\left(\ddot{z}-\ddot{z}_{srr}\right. \right.\right.\right.\nonumber\\ & \left.+ \frac{1}{2}\left({-}l_{x}\ddot{\psi } {cos} \left(\psi \right)+l_{x}\dot{\psi }^{2} {sin} \left(\psi \right)+l\left(\ddot{\beta }+\ddot{\psi }\right) {cos} \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2} {sin} \left(\beta +\psi \right)\right)\right. \nonumber\\ & \left. l\,{cos} \left(\beta +\psi \right)-2\left(\ddot{x}-\ddot{x}_{slr}-a\ddot{\theta }_{lr}+\frac{1}{2}\left({-}\ddot{\theta }l_{y}+l_{x}\ddot{\psi } {sin} \left(\psi \right)+l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)\right)-l\left(\ddot{\beta }+\ddot{\psi }\right) \right.\right.\nonumber\\ & \left.\left.{sin} \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2} {cos} \left(\beta +\psi \right)\right)l {sin} \left(\beta +\psi \right) + \left(\ddot{z}-\ddot{z}_{slr}+\frac{1}{2}\left({-}l_{x}\ddot{\psi } {cos} \left(\psi \right) \right.\right.\right.\nonumber\\& \left.\left.\left.+ l_{x}\dot{\psi }^{2} {sin} \left(\psi \right)+l\left(\ddot{\beta }+\ddot{\psi }\right) {cos} \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2} {sin} \left(\beta +\psi \right)\right)l\,{cos} \left(\beta +\psi \right)\right)\right)\end{align}

Torque of right front wheel motor:

(21) \begin{align}\tau _{\theta rf} = &\, I_{o}\ddot{\theta }_{rf}-m_{l}a(\ddot{x}-\ddot{x}_{srf}-a\ddot{\theta }_{rf}+\frac{1}{2}(\ddot{\theta }l_{y}-l_{x}\ddot{\psi }\sin \left(\psi \right)-l_{x}\dot{\psi }^{2}\cos \left(\psi \right))-l\left(\ddot{\alpha }+\ddot{\psi }\right) \nonumber\\ & \sin \left(\alpha +\psi \right)-l\left(\dot{\alpha }+\dot{\psi }\right)^{2}\cos \left(\alpha +\psi \right))\end{align}

Torque of left front wheel motor:

(22) \begin{align}\tau _{\theta lf} = & I_{o}\ddot{\theta }_{lf}-m_{l}a(\ddot{x}-\ddot{x}_{slf}-a\ddot{\theta }_{lf}+\frac{1}{2}({-}\ddot{\theta }l_{y}-l_{x}\ddot{\psi }\sin \left(\psi \right)-l_{x}\dot{\psi }^{2}\cos \left(\psi \right))-l\left(\ddot{\alpha }+\ddot{\psi }\right) \nonumber\\ & \sin \left(\alpha +\psi \right)-l\left(\dot{\alpha }+\dot{\psi }\right)^{2}\cos \left(\alpha +\psi \right))\end{align}

Torque of right rear wheel motor:

(23) \begin{align}\tau _{\theta rr} = & I_{o}\ddot{\theta }_{rr}-m_{l}a(\ddot{x}-\ddot{x}_{srr}-a\ddot{\theta }_{rr}+\frac{1}{2}(\ddot{\theta }l_{y}+l_{x}\ddot{\psi }\sin \left(\psi \right)+l_{x}\dot{\psi }^{2}\cos \left(\psi \right))-l\left(\ddot{\beta }+\ddot{\psi }\right)\nonumber \\ & \sin \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2}\cos \left(\beta +\psi \right))\end{align}

Torque of left rear wheel motor:

(24) \begin{align}\tau _{\theta lr} = & I_{o}\ddot{\theta }_{lr}-m_{l}a(\ddot{x}-\ddot{x}_{slr}-a\ddot{\theta }_{lr}+\frac{1}{2}({-}\ddot{\theta }l_{y}+l_{x}\ddot{\psi }\sin \left(\psi \right)+l_{x}\dot{\psi }^{2}\cos \left(\psi \right))-l\left(\ddot{\beta }+\ddot{\psi }\right) \nonumber \\ & \sin \left(\beta +\psi \right)-l\left(\dot{\beta }+\dot{\psi }\right)^{2}\cos \left(\beta +\psi \right))\end{align}

Longitudinal friction force applied to the right front wheel:

(25) \begin{align}& F_{xsrf} = \frac{1}{2}m_{l}\left({-}2\ddot{x}+2\ddot{x}_{srf}+2a\ddot{\theta }_{rf}-\ddot{\theta }l_{y}+l_{x}\ddot{\psi } {sin} \left(\psi \right)+l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)+2l\left(\ddot{\alpha }+\ddot{\psi }\right)\right.\nonumber\\ & \left.{sin} \left(\alpha +\psi \right)+2l\left(\dot{\alpha }+\dot{\psi }\right)^{2} {cos} \left(\alpha +\psi \right)\right) \nonumber\\ & + \frac{\mu _{dx}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)-l\,{cos} \left(\beta +\psi \right)\right)+ m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}\left(l\,{cos} \left(\alpha +\psi \right)-3l\,{cos} \left(\beta +\psi \right)\right)\right)\right)}{\left(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)\right)}.\end{align}

Longitudinal friction force applied to the left front wheel:

(26) \begin{align} & F_{xslf} = \frac{1}{2}m_{l}\left({-}2\ddot{x}+2\ddot{x}_{slf}+2a\ddot{\theta }_{lf}+\ddot{\theta }l_{y}+l_{x}\ddot{\psi } {sin} \left(\psi \right)+l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)+2l\left(\ddot{\alpha }+\ddot{\psi }\right)\right.\nonumber\\ & \left. {sin} \left(\alpha +\psi \right)+2l\left(\dot{\alpha }+\dot{\psi }\right)^{2} {cos} \left(\alpha +\psi \right)\right) \nonumber\\ & +\frac{\mu _{dx}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)-l\,{cos} \left(\beta +\psi \right)\right)+m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}\left(l\,{cos} \left(\alpha +\psi \right)-3l\,{cos} \left(\beta +\psi \right)\right)\right)\right)}{\left(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)\right)}.\end{align}

Longitudinal friction force applied to the right rear wheel:

(27) \begin{align}& F_{xsrr} = \frac{1}{2}m_{l}\left({-}2\ddot{x}+2\ddot{x}_{srr}+2a\ddot{\theta }_{rr}+\ddot{\theta }l_{y}-l_{x}\ddot{\psi } {sin} \left(\psi \right)-l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)+2l\left(\ddot{\beta }+\ddot{\psi }\right) {sin} \left(\beta +\psi \right)\right.\nonumber \\& \left.+ 2l\left(\dot{\beta }+\dot{\psi }\right)^{2} {cos} \left(\beta +\psi \right)\right) \nonumber\\ & +\frac{\mu _{dx}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)+l\,{cos} \left(\alpha +\psi \right)\right)+m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}\left(3l\,{cos} \left(\alpha +\psi \right)-l\,{cos} \left(\beta +\psi \right)\right)\right)\right)}{\left(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)\right)}.\end{align}

Longitudinal friction force applied to the left rear wheel:

(28) \begin{align}& F_{xslr} = \frac{1}{2}m_{l}\left({-}2\ddot{x}+2\ddot{x}_{slr}+2a\ddot{\theta }_{lr}+\ddot{\theta }l_{y}-l_{x}\ddot{\psi } {sin} \left(\psi \right)-l_{x}\dot{\psi }^{2} {cos} \left(\psi \right)+2l\left(\ddot{\beta }+\ddot{\psi }\right) {sin} \left(\beta +\psi \right) \right. \nonumber\\ & \left.+ 2l\left(\dot{\beta }+\dot{\psi }\right)^{2} {cos} \left(\beta +\psi \right)\right) \nonumber\\ & + \frac{\mu _{dx}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)+l\,{cos} \left(\alpha +\psi \right)\right)+m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}\left(3l\,{cos} \left(\alpha +\psi \right)-l\,{cos} \left(\beta +\psi \right)\right)\right)\right)}{\left(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)\right)}.\end{align}

Lateral friction force applied to the front wheels:

(29) \begin{align}F_{ysf}=&\, 2\mu _{dy}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)-l\,{cos} \left(\beta +\psi \right)\right)+m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}\left(l\,{cos} \left(\alpha +\psi \right)\right.\right.\right.\nonumber\\&\left.\left. \left. -3l\,{cos} \left(\beta +\psi \right)\right)\vphantom{\frac{1}{2}}\right)\right)/(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)).\end{align}

Lateral friction force applied to the rear wheels:

(30) \begin{align}F_{ysr}=& \,2\mu _{dy}\left(\frac{1}{2}M_{G}g\left(\frac{1}{2}l_{x}\, {cos} \left(\psi \right)+l\,{cos} \left(\alpha +\psi \right)\right)+m_{l}g\left(l_{x}\, {cos} \left(\psi \right)+\frac{1}{2}(3l\,{cos} \left(\alpha +\psi \right)\right.\right.\nonumber\\ &\left. \left.\left.- l\,{cos} \left(\beta +\psi \right)\right)\vphantom{\frac{1}{2}}\right)\right)/(l_{x}\, {cos} \left(\psi \right)+l\left( {cos} \left(\alpha +\psi \right)- {cos} \left(\beta +\psi \right)\right)).\end{align}

4.1. First test

As shown in Fig. 6, in the first test, the robot is in accelerating motion, with the back legs rotating from angles of 30 to 150 degrees and the front legs moving with a fixed angle of 30 degrees. The wheel speeds are set at an initial speed of zero, reaching a speed of $18.32\,rev/min$ with an acceleration of 214.58 $rev/min^{2}$ for the front wheels. The legs have a constant speed of $3.44\,rev/min$ . During this test, the wheels complete four rotations.

Figure 6.

Robot’s motion in first test.

Fig. 7 shows the rotation of the robot’s joint motors, with the back wheels (motors 5 and 8) in free mode and the front legs (motors 2 and 3) in fixed mode. The outcomes of the robot’s behavior in the simulation environment closely align with the experimental data acquired from the encoder feedback of the motors. However, a slight variance in the acceleration of the motors has resulted in discrepancies in the initial movement results.

Fig. 8 shows the rotation speed of the joint motors of the robot. In the Webots simulator for test simulation and in experimental test, the back wheels are free-spinning. Due to the free movement of the back wheels (motors 5 and 8), their rotational speed accelerates with the movement of the body. Additionally, the speed of the back legs is constant in the Webots and experimental test. As the legs complete their rotation, the speed reaches zero (motors 1 and 4).

Figure 7.

Rotation of the robot’s joint motors, (a) legs (b) wheels.

Figure 8.

Rotational speed of the robot’s joint motors, (a) leg motors and (b) wheel motors.

Fig. 9 shows the torque of the robot’s joint motors. By considering the gravitational force, the mass of the robot, the inertia of the components, and the forces between the wheels and the ground surface, the torque applied to the motors can be determined. In the Webots software, the motors are designed to be close to the real model, which is why the output torque feedback of the motors is not as sharp and smooth. Also, taking into account the ground conditions and the forces acting on the wheels from the surface, a partial amount of torque is applied to the wheel motors, resulting in slightly different results from theoretical expectations. Additionally, a small amount of torque is applied to the back wheels due to changes in the angle of the legs and surface conditions. The results obtained from the motor torque show a consistent trend, decreasing as the robot body stabilizes.

Figure 9.

Torque of the robot’s joint motors, (a) leg motors and (b) wheel motors.

In Fig. 10, the position and linear velocity of the robot’s body in the direction of the specified coordinate axes are shown. The simulation outcomes in Webots, along with theoretical simulations conducted in MATLAB, exhibit a motion characterized by a consistent slip coefficient. However, in the experimental test, due to existing slippage, the behavior of the robot differs from theoretical results. Machine vision has effectively estimated the actual path of the robot’s movement, and the experimental test data and machine vision are consistent with each other.

Figure 10.

Position and rotation angle of the robot, (a) linear position and (b) rotational position.

Fig. 11 shows the displacement of the robot on a two-dimensional plane. This image simulates observing the robot’s center from above perpendicularly. The results indicate that the lateral displacement of the robot, especially at the beginning of the movement when the wheels slip, causes the robot’s center to deviate from the straight path.

Figure 11.

The path traveled by the robot in first test.

4.2. Second test

As shown in Fig. 12, in this test, the robot moves with an accelerating motion by changing the angle of its front legs from 30 to 150 degrees and its back legs from 150 to 30 degrees. This test was started with an initial speed of zero and reached a speed of 18.32 rev/min with an acceleration of 214.58 rev/min ^ 2 on the front wheels of the robot, while the front and back legs moved at a constant speed of 3.44 rev/min. The wheels in this test rotated four times.

Figure 12.

Robot’s motion in second test.

Fig. 13 shows the rotation angle of the robot joint motors. The experimental test results acquired from the encoder feedback of the motors align well with the simulation outcomes and exhibit congruent movement behaviors. The rotation profile differs slightly between simulation in Webots and experimental test due to differences in applying acceleration to the motors of the robot.

Figure 13.

Rotation angle of robot joint motors, (a) leg motors and (b) wheel motors.

Fig. 14 shows the rotation speed of the robot joint motors. In the Webots simulation for testing, a constant speed is given to the robot joints. However, in the experimental test, the joints start moving with acceleration as the motors are initially stationary to reach the ideal input as given. The software values in Webots are considered the set speed. The motor PID coefficients are very effective in changing the rotation profile and speed, affecting the motors′ convergence speed with the input command. As the legs complete their rotation, the speed of motors drops to zero (motors 1 to 4). The back wheels are free-spinning. Because of this free movement, the rotational speed of the back wheels increases with the motion of the body. When the legs complete their movement, the speed of motors reaches zero (motors 5 and 8).

Figure 14.

Rotation speed of robot joint motors, (a) leg motors and (b) wheel motors.

Fig. 15 shows the torque of the robot joints motors. In Webots software, the motors are designed closely to the real model, which is why the output torque feedback of the motors is not as sharp and smooth as the theoretical results. Due to the complexity of this test and differences in the motor’s PID coefficients that determine the importance of input torque, and not seeing the applied torques from other factors, the alignment of theoretical relationships with simulation has been slightly reduced to experimental testing.

Figure 15.

Torque of robot joint motors, (a) leg motors and (b) wheel motors.

In Fig. 16, the position and linear velocity of the robot body in the direction of specified coordinate axes are shown. The simulation outcomes within Webots, alongside the theoretical frameworks, both suggest a motion characterized by a consistent slip coefficient. However, disparities in the applied acceleration procedures between the two methods account for the variations observed. The machine vision results are close to the experimental test considering the existing slip, and the robot’s behavior differs from theoretical results. Image processing has successfully estimated the robot’s actual movement path, and the experimental test data and image processing are consistent with each other. Furthermore, the inertial sensor and image processing worked well in determining the robot’s position in real conditions and were able to estimate the deviation of the robot from the main path under slip conditions and measure changes in the body’s height with good accuracy consistent with simulation.

Figure 16.

Position and rotation angle of the robot, (a) linear position and (b) rotational position.

Fig. 17 shows the displacement of the robot on a two-dimensional plane. This image depicts the overall movement of the robot on the plane and its trajectory. The ideal movement state of the robot is straight motion without lateral displacement; however, the results indicate that the lateral displacement of the robot at the beginning of the movement, when the wheels slip, causes the robot to deviate from the main path. At the end of the path, due to the body’s deviation from the path, significant lateral displacement occurs.

Figure 17.

The path traveled by the robot in second test.