To demonstrate the advantages of the proposed HDSA optimization algorithm, the results of the HDSA tests are shown in this section. The theory of HDSA is discussed in detail in the section entitled “RBFNN-Based Active Fault-Tolerant Control Algorithm”. The effectiveness of the proposed optimization algorithm was evaluated by comparing the test results of HDSA with other popular optimization algorithms, such as particle swarm optimization (PSO) (Song and Gu, 2004), the sine/cosine algorithm (SCA) (Mirjalili, 2016), the gray wolf optimizer (GWO) (Mirjalili et al., 2014), the firefly algorithm (FA) (Fister et al., 2013), and the Harris hawk optimizer (HHO) (Heidari et al., 2019). Twenty standard test functions were used for evaluation, which are presented in Tables 5–7 (included in the Simulation Results section).

The number of populations was pop = 100 and the maximum number of iterations was M = 100. The average fitness over 30 independent runs was considered as the optimization result. The convergence characteristics of the six algorithms in the single-peak function test are depicted in Figure 1, while Figure 2 illustrates the convergence characteristics in the multi-peak function test. Furthermore, Figure 3 demonstrates the convergence characteristics of the six algorithms on fixed-dimensional multi-peak functions. The test results of the six algorithms, based on 30 independent runs, are summarized in Tables 1, 2. In Tables 1, 2, purple indicates the optimal value of the test functions, pink indicates the mean value of the test functions, and white indicates the mean squared deviation of the test functions.

The results of the single-peak functions F1–F7 test results are presented in Tables 1, 2. In these tests, the mean and optimal values obtained by HDSA in F1–F5 are both 0, indicating that HDSA achieves the highest accuracy among the six algorithms. Although the accuracy of HDSA is slightly inferior to HHO in the F6–F7 test functions, it still outshines SCA, PSO, GWO, and FA. HDSA has a standard deviation of 0 in tests F1–F5, suggesting that HDSA is the most stable algorithm. Although its stability is slightly lower than HHO in tests F6–F7, it still outperforms the other four methods. Convergence speed is depicted in Figure 2. HDSA has a significantly faster convergence speed compared with the other five algorithms, but its convergence accuracy in the F6–F7 tests is lower than that of HHO.

The test results for the multi-peak functions F8–F13 are presented in Tables 1, 2. In the tests from F9 to F13, HDSA exhibits significantly better stability and convergence accuracy compared with the other five algorithms. It achieves higher accuracy and the smallest standard deviation. As depicted in Figure 3, except for the F8 test function, HDSA showcases the fastest convergence speed and highest convergence accuracy among the algorithms.

The results of the fixed dimensional multi-peak functions F14–F20 test results are shown in Tables 1, 2. In the F14 test, SCA has the best optimal and average accuracy, while HDSA exhibits slightly lower average accuracy and stability compared with SCA, PSO, and HHO. However, HDSA still manages to find the optimal solution in 30 runs. In the F15–F18 test results, HDSA, SCA, GWO, and HHO perform closely, with good stability and accuracy. In the F19–F20 tests, HDSA outperforms the other five algorithms significantly in terms of accuracy and stability. As shown in Figure 3, HDSA exhibits the fastest convergence speed among the other test functions, except for F15, F17, and F18. In the F15 test, HDSA is only slightly slower than HHO, while in the F17 and F18 tests, HDSA converges slightly slower than FA.

Before discussing the mathematical model of the CDR, the following assumptions are made: Assumption 1: The center of gravity and the geometric center of the robot body coincide. Assumption 2: The motor output torque meets the actual performance requirements of the robot during ground and water motion. Assumption 3: The robot's vertical swing, horizontal rocking, and longitudinal rocking during its movement on the water surface are ignored. Assumption 4: The motion of the robot on the ground is purely rolling, without any sliding motion.

The CDR designed in this study can be seen as a combination of a quadrotor UAV and a WMR. Figure 4A shows the robot moves on the ground. Figure 4B shows the robot moves on the water surface by webbed plates. Figure 4C shows the robot moves on the water surface by propllers. The robot moves in the air in a similar way to the quadrotor UAV as shown in Figures 4D, E. Figure 4F shows the structure of the robot, where webbed plates are mounted on the wheels. These webbed plates generate traction and rotational torque on the water surface by interacting with the water. However, as this paper focuses primarily on the FTC algorithm of the robot on the ground and on the water surface, the discussion does not explore the robot's aerial motion in detail.

The robot in the inertial frame and in the body frame is shown in Figure 5.

In Figure 5, d is the distance from the geometric center of the robot O_b to the mass center of the robot. b is the axis radius and r is the wheel radius. ω_l, ω_r are the angular velocities of the left and right wheels. ψ is the angle between the robot body coordinate system b and the inertial coordinate system A, and ψ is the yaw angle of the robot. The kinematic model of the robot on the ground and water surface can be represented as (Liu et al., 2020):

where q=[xyψ] represents the position and orientation of the robot in the inertial frame, while η=[uvr] is used to denote the longitudinal velocity, lateral velocity, and yaw angular velocity in the body frame. The coordinate conversion matrix is denoted by R, where R=[cosψsinψ0-sinψcosψ0001]. The dynamics model of the robot's motion on the ground can be expressed as

The matrices M are symmetric positive definite inertia matrices, while C_m represents the centripetal and Coriolis matrix. The term F(q˙) denotes mechanical friction, while τ_d is used to represent external disturbances. The input transformation matrices are denoted as B(q). Furthermore, the robot drive motors in the left and right wheel output torque are represented by τ=[τlτr]T.

The mass of the robot is represented by m. The I is a scalar quantity and represents the rotational inertia of the robot as it rotates in the X-Y plane. The angular velocity of the robot is assumed to vary smoothly, so that ψ˙≈0. According to assumption 1, the Coriolis matrix can be assumed to be negligible, resulting in C_m ≈ 0. According to assumption 1, d = 0, so the matrix M(q)=diag[mmI]. Based on these assumptions, the dynamics model of the robot on the ground can be rewritten as follows:

where C̄=R- 1CmṘ, M̄=R-1MR, B̄=R-1B. F̄(q˙)=[fufvfr]Tis the mechanical friction and τ̄d=[dudvdr]T is the external disturbance. Rewriting 3 into algebraic form can be expressed as:

The traction force is represented by F_u, while T_r represents the torque. To model the dynamics of the robot on the water surface, we can refer to the USV dynamics model (Chen et al., 2019), which can be expressed as follows

M_w is the inertia matrix. The traction force and torque of the robot at the water surface are τw=[Fu0Tr]T. τdw=[duwdvwdrw]Tis the lumped disturbance and Fw(η)=[fuwfvwfrw] is the water resistance.

The disturbances are represented by τ_dw. On the other hand, D_w(η) represents the water resistance. The Coriolis force matrix can also be neglected according to Assumption 1 and Assumption 3, so C_w(q, η) ≈ 0. The elements of the non-diagonal matrix in matrix D_w(η) and matrix M_w are small and can be neglected. This model simplification approach is also more common (Liao et al., 2016; Wang et al., 2019b; Deng et al., 2020), where m11=m-Xu˙, m22=m-Yv˙, and m₃₃ = I_z−N_ṙ are the inertia parameters of the three axes and Xu˙, Yv˙, and N_ṙ are the additional inertia parameters due to the wet water of the robot shell and the viscosity of the water. The dynamics model of the robot on the water surface can be expressed as:

X_u, X_|u|u, Y_u, Y_|v|v, and N_ω, N_|ω|ω are the resistance coefficients. The resistance of the robot moving on the water surface can be approximated as a quadratic function of the velocity and angular velocity.

The mathematical model should be rewritten into a form that better suits the needs of the subsequent controller design. The dynamics model of the robot's motion on the ground is rewritten according to 4 as

Where d_ug is the lumped disturbance and dug≤d̄ug, d̄ug is the upper limit of the total disturbances. d_rg is the lumped disturbance and drg≤d̄rg, d̄rg is the upper limit of the total disturbances. The dynamics model of the robot on the water surface is

where F_uc is the desired tractive force and F_uc = F_u represents no force loss. ξu∈[01) is the force loss parameter. Δ_F is the force disturbance due to mass change. d_uw is a lumped disturbance, duw≤d̄uw. d̄uw is the upper bound of d_uw. D_uw is the uncertainty term when the robot moves on the water surface due to changes in system parameters, water resistance, and driver faults. T_rc is the desired torque and T_rc = T_r represents no force loss. ξr∈[01) is the power loss parameter. Δ_T is the torque disturbance due to the change of inertia parameter. d_rw is a lumped disturbance, drw≤d̄rw. d̄rw is the upper bound of d_rw. D_rw is the uncertainty term due to changes in system parameters, water resistance, and driver faults during robot rotation on the water surface.

Both the yaw control and the linear velocity control of the robot are essentially single-input single-output (SISO) second-order non-linear affine systems. Without loss of generality, a second-order non-linear affine SISO system with drive faults can be expressed as:

u_c is unconstrained control input, u_a is the drive bias, ξ is the power loss parameter, ξ∈[01), 0 represents no power loss, and 1 represents a complete loss of efficiency. D = −g(x)ξu_c + u_a is the uncertainty term due to the driver fault. The disturbance d has a well-defined upper limit and |d|≤d̄. x₁, x₂ are system states. f(x) is the system function and g(x) is the input function. Owing to the physical constraints of the controlled object, the control input is subject to saturation:

u_max is the physical constraint. To make the control input smoother, the cutoff function is usually replaced by a saturation function, such as tanh.

where u_con is the constrained control input and u_f is a function of u_c. Thus, the control objective is to design the constrained control law u_con so that it satisfies the control requirements even in the presence of drive faults and external disturbances in the controlled object. The steps for designing an AFT controller are the following:

Step 1: Define the state error e₁ = x_1d − x₁. Establish the Lyapunov function V1=12e12. Taking the derivative of V₁ with respect to the time t gives

Define the virtual state α_x = k₁e₁ + ẋ_1d as the desired input of the next step. If x₂ can follow α_x, V˙1=-k1e12. So, the next step of the control law must ensure that α_x − x₂ = 0. α_x is the next desired state x_2d.

Step 2: Define the state error e₂ = x_2d − x₂, and the fast NTSMC is designed as

where α and β are positive adjustable parameters and λ is a positive odd number. The sliding mode convergence law is

where k₁, k₂, and γ₁ are positive adjustable parameters. sgn is the symbolic function. The derivation of 13 yields:

where

The controller law can be designed as follows:

In 17, the uncertain term due to drive faults D is known. Establishing the Lyapunov function V2=12S2, the derivative of V₂ yields

Bringing 17 into 18 yields

When k3>d̄/|S|γ1, k3|S|γ1-d̄=ε, ε > 0, thus:

where α₁ = 2k₂, 0<β1<2ε.

Lemma 1 [44] (Jiang and Lin, 2020): Consider a smooth positive definite V(x), x ∈ R_n. Suppose that real numbers p₁ ∈ (0, 1), α > 0, and β > 0 exist such that V(x)<-αV(x)p1-βV(x). Then, an area U₀ ∈ R_n exists, such that any V(x) starting from U₀ can reach V(x) = 0 in finite time T_v, which is expressed as Tv≤1β(1-p1)ln (V1-p1(x0)+αα).

According to lemma 1, V₂ can converge to 0 in finite time. In the above discussion, the uncertainty term D is assumed to be known, but the actual uncertain term D is unknown. As RBFNN can approximate arbitrary uncertain non-linear functions and does not depend on a mathematical model, it is more suitable for estimating stochastic uncertain terms. Therefore, optimal neural network weights w^* must exist such that D=ε0+w*Th, ε₀ is the estimated residual and h is the neuron. w~=ŵ-w*, ŵ is an estimate of w^* and w^* is a constant, so w~˙=ŵ˙. Rewrite 9 as:

Step 3: Establish the Lyapunov function V₃ as

The derivation of formula 22 yields

The control law is designed to

Bringing formula 24 into 23 yields

where ε₁ = d + ε₀, the upper limit of the estimation error of the neural network is ε̄0 . ε̄0≥ε0, d̄≥d, so that ε1≤d̄+ε̄0=ε̄1 . The update law of the RBFNN weights is designed as

Bringing 26 into 25 yields

when k3>ε̄/|S| γ1, k3|S| γ1-ε̄=ε2, where ε₂ > 0, thus:

According to lemma 1, V₂ can converge to 0 in finite time.

The control input u_c in formula 24 is the unconstrained, to prevent the control input saturation, define u_d = u_c, where u_d is the desired value in the next step, and the state error e₃ = u_d−u_con. u_con satisfies the constrained control input of the saturation function tanh; therefore, parameter u_f must exist, such that u_con = u_maxtanh(u_f/u_max), where u_max is the maximum input.

Step 4: Establish the Lyapunov function V4=12e32 and derive V₃ and bring it into 29 to obtain:

u˙f is designed as

where δ = |u_f|−2u_max, Δ is a smaller normal value. γ₂ ∈ (0, 1). The convergence of the controller is discussed in the following cases. When δ ≥ Δ, substituting 31 into 30 yields

where 0<α2<2(γ2+1)/2, 2k₃ = β₂. According to Lemma 1, V₄ can converge to 0 in finite time. When δ < Δ, substituting 31 into 30 yields

where α₃ = (γ₂ + 1)/2, c=|δ| γ22(γ2+1)/2/(1-tanh2(uc/umax)), and tanh(u_c/u_max) < 1, so c > 0. According to Lemma 2, V₄ can converge in finite time.

Lemma 2: Chu et al. (2022) Suppose that there is a positive definite continuous Lyapunov function V(x, t) defined on U1×R+, where U₁ ⊆ U ⊆ R_n. R_n is a neighborhood of the origin, and V(x,t)≤-cVα(x,t),∀x∈U1\{0}, where c > 0, 0 < α < 1. Then, the origin of the system is locally finite time stable. The settling time T≤V1-α(x(t0),t0)/c(1-α) satisfies for a given initial condition x(t₀) ∈ U₁.

The human decision search algorithm (HDSA) is a swarm optimization technique that mimics the decision-making process of a human crowd. In many post-apocalyptic survival games or films, the strong group consciousness of humans is often portrayed, but the importance of individual consciousness is also emphasized. In human groups, a small group of individuals called decision-makers make the final decisions based on their experience and personal status. However, the decision of the decision-maker is not necessarily optimal. When the number of individuals in the group is small, it is important to involve more people in the decision-making process to guide the development of the group and to avoid the excessive impact of individual decisions on the group. However, when the number of individuals in the group is large, the proportion of decision-makers should be reduced and only a few elite individuals should be selected to determine the development of the group. This is because too many people involved in the decision-making process may take more time, and the experience of ordinary people may not be as good as that of elite individuals. Because people have emotions, they can think both rationally and emotionally when dealing with problems, and these two opposing ways of thinking must coexist.

Apart from the decision-makers, the rest of the human population is referred to as the executors, consisting of individuals who have no or less ability to make decisions. They carry out the optimal decisions made by the decision-makers. However, individuals among the executors who have some decision-making ability should be encouraged to seek more humane decisions based on the optimal decisions. These decisions should become more adapted to the current environment over time. The number of decision-makers is fixed, and elite individuals in the human population will always be selected as decision-makers. Over time, any individual has the potential to become a decision-maker, and the current decision-maker may become an executor.

In a human population, there are always individuals who question the current decision or believe they have a better one, including the decision-makers themselves. These individuals are known as adventurers, and their numbers and identities are random, making them a source of uncertainty within the population. Although adventurers can lead people to a better life, they can also lead them to disaster. Adventurers, on the other hand, inherit the current optimal choices of the human population and take them into account when making decisions. However, more adventurous individuals will also seek out possible optimal decisions based on their own state. To avoid harming the human population, adventurers must consider whether the decisions they make are more beneficial to their own survival. Additionally, there is a chance that an adventurer will become a decision-maker if they come up with a better or suboptimal decision. Based on the above analysis, the proposed algorithm for optimizing the human decision population consists of three main components: decision updating for decision-makers, decision updating for executors, and decision updating for adventurers.

According to the control algorithm in the “RBFNN-Based Active Fault-Tolerant Control Algorithm” section, the AFTC is used to design controllers in this section to follow the desired yaw angle ψ_d and desired linear velocity v_d. The robot linear velocity sliding mode surface is: Sv=αvev+βvevλv, where e_v = v_d−v. The sliding mode convergence law is Ṡv=-k2vSv-k3v|S| γ1vsgn(Sv).

The proof of convergence for the velocity controller is similar to that for the general-purpose controller in the “RBFNN-Based Active Fault-Tolerant Control Algorithm" section. The unconstrained control law is designed as

The anti-input saturation controller of linear velocity is designed as

Where e_F = F_uc−F_ucon.

The yaw angle controller is ωd=kψeψ+ψ˙d, where e_ψ = ψ_d − ψ. The yaw angle sliding mode surface is Sω=eω+αψeψ+βψeψλψ. The sliding mode convergence law is Ṡω=-k2ωSω-k3ω|Sω|γ1ωsgn(Sω).

The unconstrained control law is designed as

The anti-input saturation controller of the yaw angle is designed as

where e_T = T_rc−T_rcon. The controller parameters are not described in this section as they have been discussed in the “RBFNN-Based Active Fault-Tolerant Control Algorithm" section.

The input to the angular velocity neural network is both the yaw error and the angular velocity error, and the output is the uncertainty term in the angular velocity control. The coordinate vector matrix of the centroids of the Gaussian basis function neurons in the angular velocity neural network is

cψ=[-1.6-0.8-0.4-0.2-0.100.10.20.40.81.6-1.6-0.8-0.4-0.2-0.100.10.20.40.81.6] 2*11

The width of the Gaussian basis function b_ψ = 0.1, i = 1⋯11.

The input to the linear velocity neural network is the velocity error and the output is the linear velocity control uncertainty term. The coordinate vector matrix of the centroids of the Gaussian basis function of the neurons in the linear velocity neural network is

cv=[-1.6-0.8-0.4-0.2-0.100.10.20.40.81.6]1*11. The width of the Gaussian basis function b_v = 0.1, i = 1⋯11.

Based on the above discussion, the proposed framework for the AFTC is shown in Figure 6.