1. Introduction

2. Literature Review

3. Dynamic Models and the Controller Design of a Wheeled Mobile Robot

3.1. Development of a Kinematic Model of a Four-Wheeled Mobile Robot for Agricultural Applications

Case study and assumptions

Assumptions in dynamic model development are principles, simplifications and approximations that make the model tractable and computationally feasible, while capturing real-world system characteristics. Common assumptions depend on the model’s nature and modelling objectives [8]. The importance of assumptions in wheeled mobile robot model design lies in their ability to simplify the problem, capture the essential characteristics, enable analysis and simulation, facilitate design decisions, assess the model’s limitations, and enable comparisons and benchmarking. Careful consideration of these assumptions is crucial for developing accurate and reliable models that can guide the design and development of effective wheeled mobile robots. The assumptions in this study were:

• linearity: assuming that the system behaves in a linear manner, where the output is directly proportional to the input, this simplifies the mathematical analysis and allows for the use of linear techniques
• time-invariance: assuming that the system’s behaviour does not change over time, this means that the system’s properties and characteristics remain constant throughout the modelling period
• homogeneity: assuming that the system’s behaviour is uniform throughout its spatial domain, this means that the system’s properties and characteristics are the same at all points in space
• determinism: assuming that the system’s behaviour is fully determined by its initial conditions and inputs, this excludes the influence of random or stochastic factors
• perfect knowledge: assuming that all relevant information about the system is known and available for modelling purposes, this includes complete knowledge of the system’s structure, parameters and boundary conditions.

Agriculture, a major economic sector, contributes 6.4 % of global productivity and employs a significant workforce. In view of concerns about food security, population growth and climate change, innovative approaches such as wheeled mobile robots are being explored. Figure 1 shows a wheeled mobile robot in use in agricultural areas. This wheeled mobile is dynamic since it observes the entire plant’s health status as it moves along.

Following the literature review, kinematical models of a wheeled mobile robot were developed (Figure 2). Based on the kinematical models of a wheeled mobile robot, state space was generated. The kinematical models of wheeled mobile robots then required regulation, and the optimal controllers were selected for this. If the gain obtained was optimal, it was directly provided to the dynamical models of the wheeled mobile robot. Subsequently, data were collected from the dynamic models of the wheeled mobile robot for analysis purposes. If the gain was not optimal, the controller design was examined again, with the tuned and relaunched system checked until the desired optimal gain was obtained.

Considering only the kinematic equation model of the SSMR (Figure 3), the following velocity constraint was obtained:

$$v_y+x_{ICR}\dot{\theta }=0$$

(1)

Where $x_{ICR}$ is inertia centre, $v_y$ is vertical velocity of the robot, and $\dot{\theta }$ is angular velocity of the mobile robot.

Equation (1) is not integrable and is consequently a nonholonomic constraint. It can therefore be rewritten in the form $\left[-\sin \theta \cos \theta x_{ICR} \right]\left[\dot{x} \dot{y } \dot{\theta }\right]=A\left(q\right)\dot{q}=0 ,$where equation (1) is used. Since the generalised velocity is always in the null space of A, the last equation is not integrable. Consequently, it describes a nonholonomic constraint that can be rewritten in the Pfaffian form:

$$\left[-sin\theta \hspace{0.17em}cos\theta \hspace{0.17em}{x}_{ICR}\right]\left[\dot{X}\dot{Y}\dot{\theta }\right]=A\left(q\right)\dot{q}=0$$

(2)

$$\dot{q}=S\left(q\right)\eta$$

(3)

Where $S\left(q\right)\eta$ is state space variable matrix.

Rearranging the kinematics models, the state space equation below was finally achieved:

$$S\left(q\right)=\left[\begin{array}{c}\cos \theta x_{ICR}\sin \theta \\ \sin \theta -x_{ICR}\cos \theta \\ \end{array}\right]$$

(4)

The input matrix is the following equation:

$$\beta =\left[\begin{array}{c}{V}_x\\ \omega \end{array}\right]=r\left[\begin{array}{c}\frac{{\omega }_L+{\omega }_R}{2}\\ \frac{{-\omega }_L+{\omega }_R}{2c}\end{array}\right]$$

(5)

Where $\beta$ is velocity vectors of both linear and angular velocity.

Then the dynamic properties of the SSMR were described, since the dynamic effects play an important role in such vehicles. This is caused by unknown lateral skidding ground interaction forces. First, the wheel forces depicted in Figure 3 were examined.

Figure 3. Forces acting on one wheel [8].

Figure 3. Forces acting on one wheel [8].

The active force Fi and reactive force $N_i$ are related to the wheel torque and gravity, respectively. It is clear that $F_i$is linearly dependent on the wheel control input ${\tau }_i$ , namely:

$$F_i=\frac{{\tau }_i}{r}$$

(6)

It was assumed that the vertical force $N_i$ acts from the surface to the wheel. Considering the four wheels of the vehicle (Figure 4) and neglecting additional dynamic properties, the following equations of equilibrium were obtained:

$$\left\{\begin{array}{c}{N}_{1}a=N_{2}b\\ N_{3}a=N_{4}b\\ {\displaystyle \sum _{i=1}^4}{N}_i=mg\end{array}\right.$$

(7)

where m denotes the vehicle mass and g is the gravity acceleration. Where $p_1,p_2,p_3,p_4$ are the position of robot wheels, $F_1,F_2,F_3,F_4$, $F_{S1},{FS}_2,F_{S3},F_{S4}$, $F_{l1},F_{l2},F_{l3},F_{l4}$, are forces on the wheels, M moment of the wheels, ${\tau }_r,{\tau }_i,$torques of the wheels $V_i$ velocities of the wheel, V velocity of the mobile robot. Since there is symmetry along the longitudinal midline, the following was obtained:

$$\left\{\begin{array}{c}{N}_1=N_4=\frac{b}{2\left(a+b\right)}mg \\ N_2=N_3=\frac{a}{2\left(a+b\right)}mg \end{array}\right.$$

(8)

It was assumed that the vector $F_{si}$results from the rolling resistant moment ${\tau }_{ri}$ and the vector $F_{li}$denotes the lateral reactive force. These reactive forces can be regarded as friction forces. However, it is important to note that friction modelling is quite complicated since it is highly nonlinear and depends on many variables. Therefore, in most cases only a simplified approximation describing the friction $F_f$as a superposition of Coulomb and viscous friction is considered. It can be written as:

$$F_f\left(\sigma \right)={\mu }_{c}Nsgn\left(\sigma \right)+{\mu }_v\sigma$$

(9)

where σ denotes the linear velocity, N is the force perpendicular to the surface, and ${\mu }_c$ and ${\mu }_v$ denote the coefficients of Coulumb and viscous friction, respectively. Since for the SSMR the velocity σ is relatively low, especially during lateral slippage, the relation ${\mu }_{c}N \gg \left|{\mu }_v\sigma \right|$ is valid, which allowed the term ${\mu }_v\sigma$ to be neglected to simplify the model. It is very important to note that the function of a kinematics system is not smooth when the velocity σ equals zero, because of the sign function sgn (σ). It is obvious that this function is not differentiable at $\sigma =0$. Since the aim was to obtain a continuous and time-differentiable model of the SSMR, the following approximation of this function was proposed:

$$\widehat{sgn}\left(\sigma \right)=\frac{2}{\pi }\arctan \left(k_s\sigma \right)$$

(10)

where $k_s\gg 1$ is a constant that determines the approximation accuracy according to the relation

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{2}{\pi }\arctan \left(k_s\sigma \right)=sgn\left(x\right)$$

(11)

Based on the previous deliberations, the friction forces for one wheel can be written as:

$$F_{li}={\mu }_{lci}mg\widehat{sgn}\left(v_{yi}\right)$$

(12)

$$F_{si}={\mu }_{sci}mg\widehat{sgn}\left(v_{xi}\right)$$

(13)

where ${\mu }_{lci}$ and ${\mu }_{sci}$ denote the coefficients of the lateral and longitudinal forces, respectively. Using the Lagrange-Euler formula with Lagrange multipliers to include the nonholonomic constraint (1), the dynamic equation of the robot can be obtained. Next, it was assumed that the potential energy of the robot $P E \left(q\right)=0$ because of the planar motion. Therefore, the Lagrangian L of the system equals the kinetic energy:

$$L\left(q,\dot{q}\right)=T\left(q,\dot{q}\right)$$

(14)

Considering the kinetic energy of the vehicle and neglecting the energy of rotating wheels, the following equation was developed:

$$T=\frac{1}{2}m{v}^{T}v+\frac{1}{2}I{\omega }^2$$

(15)

where m denotes the mass of the robot and I is the moment of inertia of the robot about the COM. For simplicity, it was assumed that the mass distribution is homogeneous. Since

$$v^{T}v=v_x^2+v_y^2={\dot{X}}^2+\dot{Y^2}$$

(16)

substituting equation (16) into eqn. (15) can be rewritten in the following form:

$$T=\frac{1}{2}m\left(\dot{X^2}+{\dot{Y}}^2\right)+\frac{1}{2}I{\dot{\theta }}^2$$

(17)

After calculating the partial derivative of kinetic energy and its time derivative, the inertial forces could be obtained as

$$\frac{d}{dt}\left(\frac{\partial E_k}{\partial \dot{q}}\right)=\left[\begin{array}{c}m\ddot{X}\\ m\ddot{Y}\\ I\ddot{\theta }\end{array}\right]=M\ddot{q}$$

(18)

where

$$M=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& m\end{array}\right]$$

(19)

Consequently, the forces that cause the dissipation of energy were considered. In accordance with Figure 2, the following resultant forces expressed in the inertial frame could be calculated:

$$F_{ry}\left(\dot{q}\right)=cos\theta \sum _{i=1}^4{F}_{si}\left(v_{xi}\right)-sin\theta \sum _{i=1}^4{F}_{li}\left(v_{yi}\right)$$

(20)

$$F_{ry}\left(\dot{q}\right)=sin\theta \sum _{i=1}^4{F}_{si}\left(v_{xi}\right)+cos\theta \sum _{i=1}^4{F}_{li}\left(v_{yi}\right)$$

(21)

The resistant moment around the centre of mass $M_r$ can be obtained as

$$\left\{\begin{array}{c}{M}_r(\dot{q)}=-a{\displaystyle \sum _{i=\mathrm{1,4}}}{F}_{li}\left(v_{yi}\right)+b{\displaystyle \sum _{i=\mathrm{2,4}}}{F}_{li}\left(v_{yi}\right)\\ +c\left[-{\displaystyle \sum _{i=\mathrm{1,2}}}{F}_{si}\left(v_{xi}\right)+{\displaystyle \sum _{i=\mathrm{3,4}}}{F}_{si}\left(v_{xi}\right)\right]\end{array}\right.$$

(22)

To define generalised resistive forces, the vector

$$R\left(\dot{q}\right)={\left[F_{rx}\left(\dot{q}\right)F_{ry}\left(\dot{q}\right)M_r\left(\dot{q}\right)\right]}^T$$

(23)

was introduced. The active forces generated by the actuators which make the robot move can be expressed in the inertial frame as follows:

$$F_x=cos\theta \sum _{i=1}^4{F}_i$$

(24)

$$F_y=sin\theta \sum _{i=1}^4{F}_i$$

(25)

The active torque around the common point was calculated as:

$$M=c\left(-F_1-F_2+F_3+F_4\right)$$

(26)

Consequently, the vector F of active forces had the following form:

$$F={\left[F_x{F}_{y}M\right]}^T$$

(27)

Using (4), (26) and (27) and assuming that the radius of each wheel is the same, this gave

$$F=\frac{1}{r}\left[\begin{array}{c}cos\theta {\displaystyle \sum _{i=1}^4}{\tau }_i \\ sin\theta {\displaystyle \sum _{i=1}^4}{\tau }_i\\ c\left({-\tau }_1-{\tau }_2+{\tau }_3+{\tau }_4\right)\end{array}\right]$$

(28)

To simplify the notation, a new torque control input τ was defined as

$$\tau =\left[\begin{array}{c}{\tau }_L\\ {\tau }_R\end{array}\right]=\left[\begin{array}{c}{\tau }_1+{\tau }_2\\ {\tau }_3+{\tau }_4\end{array}\right]$$

(29)

where ${\tau }_L$ and ${\tau }_R$ denote the torques produced by the wheels on the left and right sides of the vehicle, respectively. Combining (26) and (27), this gives

$$F=B\left(q\right)\tau$$

(30)

where B is the input transformation matrix defined as:

$$B\left(q\right)=\frac{1}{2}\left[\begin{array}{c}cos\theta cos\theta \\ sin\theta sin\theta \\ -c c\end{array}\right]$$

(31)

Next, using (1), (23) and (30), the following dynamic model was obtained:

$$M\left(q\right)\ddot{q}+R\left(\dot{q}\right)=B\left(q\right)\tau$$

(32)

It should be noted that (30) describes the dynamics of a free body only and does not include the nonholonomic constraint (2). Therefore, a constraint had to be imposed on (30). To this end, a vector of Lagrange multipliers, λ, was introduced as follows:

$$M\left(q\right)\ddot{q}+R\left(\dot{q}\right)=B\left(q\right)\tau +A^T\left(q\right)\lambda$$

(33)

For control purposes it would be more suitable to express (29) in terms of the internal velocity vector η. Therefore, (33) was multiplied from the left by $S^T\left(q\right)$, which resulted in:

$$\left\{\begin{array}{c}{S}^T\left(q\right)M\left(q\right)\ddot{q}+S^T\left(q\right)R\left(\dot{q}\right)\\ =S{\left(q\right)}^{T}B\left(q\right)\tau +{\mathrm{S}}^{\mathrm{T}}\left(q\right)A^T\left(q\right)\lambda \end{array}\right.$$

(34)

After taking the time derivative of (3), this gave:

$$\ddot{q}=\dot{S}\left(q\right)\eta +S\left(q\right)\dot{\eta }$$

(35)

Next, using (33) and (3) in (31), the dynamic equations became

$$\overline{M}\dot{\eta }+\overline{C}\eta +\overline{R}=\overline{B}\tau$$

(36)

where

$$\overline{C}=S^{T}M\dot{S}=m{x}_{ICR}\left|\begin{array}{cc}0& \dot{\theta }\\ \dot{\theta }& \dot{x_{ICR}}\end{array}\right|$$

(37)

$$\overline{M}=S^{T}MS=\left[\begin{array}{cc}m& 0\\ 0& m{x}_{ICR}^2+I\end{array}\right]$$

(38)

$$\overline{R}=S^{T}R=\left[\begin{array}{c}{F}_{rx}\left(\dot{q}\right)\\ x_{ICR}{F}_{ry}\left(\dot{q}\right)+M_r\end{array}\right]$$

(39)

$$\overline{B}=S^{T}B=\frac{1}{2}\left[\begin{array}{cc}1& 1\\ -c& c\end{array}\right]$$

(40)

4. Results and Discussion

5. Conclusions

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