# Predictive Control for Small Unmanned Ground Vehicles via a Multi-Dimensional Taylor Network
## 4 MTN Predictive Control Scheme with the Differential Term
### Scheme of Predictive Control with the Differential Term Based on the MTN Model  
An improved predictive control scheme on the basis of the MTN is proposed for tracking control of SUGVs. First, a predictive model is constructed as follows: (1) the MTN is used as a predictive model to construct a SUGV model; and (2) the BP algorithm is used as its learning algorithm. Second, a predictive control law is designed to control the SUGV. The frame diagram of the improved MTN predictive control system is shown in Figure 3. The front wheel steering angle \(\delta_{f}\) is used as the control input, which is equivalent to the \(u(k)\) mentioned above. The yaw velocity \(w\) is used as the system output, which is equivalent to the \(y(k)\) mentioned above. The predictive model output \(\widehat{w}\) is equivalent to the \(\hat{y}(k)\) mentioned above. The desired yaw velocity \(w_{des}\) is used as the system input, which is equivalent to the \(r(k)\) mentioned above. The specific steps are as shown in Algorithm 1. ``` Step 1: Construct MTN predictive model; (1) Initialize system input and output; (2) Asceratin structure of MTN predictive model; (3) Update \(\mathbf{w}_{\mathbb{P}}(k)\) with Equation (12); Step 2: Construct MTN predictive control law; (1) Construct the quadratic performance index, shown as (13); (2) Update \(u(k)\) with Equation (14); Step 3: Construct an improved predictive control law to control SUGV based on MTN predictive control law; (1) Construct the quadratic performance index with differential term, shown as (15); (2) Update the \(u(k)\) with Equation (17); Step 4. Implement the tracking control for the SUGV. ## 5 Simulation Results and Discussion

### Simulation Results

SUGVs have a wide range of applications, such as delivery, sanitation, and mining. A tracking control experiment on a SUGV is described here [44; 45]. Moreover, the traditional MTN predictive control scheme [24] and the recurrent radial basis (RBF) model predictive control scheme [46] are used for comparison. According to the vehicle dynamics model, as shown in (7), the tracking effect is demonstrated by accurately tracking the desired yaw velocity signal. The control input and output variables are the front wheel steering angle \(\delta_{f}\) and the yaw velocity \(w\), respectively. The mathematical relationship between the desired path curve \(y=f(x)\) and the desired yaw velocity signal is as follows [44; 45]:

\[w_{des}=v_{des}k_{des} \tag{23}\]

\[k_{des}=\frac{d^{2}y/dx^{2}}{\left(1+\left(dy/dx\right)^{2}\right)^{3/2}} \tag{24}\]

where \(w_{des}\) is the desired yaw velocity, \(v_{des}\) is the desired speed, and \(k_{des}\) represents the curvature of the desired road. Specific parameters of the SUGV were set as follows: \(M=0.1\) kg, \(l_{f}=0.1\) m, \(l_{r}=0.1\) m, \(l_{z}=0.4\) kg/m\({}^{2}\), \(v_{x}=1\) m/s, \(L=0.6\) m, \(C_{f}=0.6\) N/rad, and \(C_{r}=0.6\) N/rad. Two cases were considered: without noise and with noise. The same parameters were adopted for each of them to verify the performance of the proposed scheme. **Remark 1**. : _In the following, D-MTNC is our proposed scheme; MTNC is the traditional MTN predictive control scheme; and RBFC is the RBF model predictive control scheme._

**Remark 2**. : _In the following figures, vdes represents the given reference signal; yMTNC is the traditional MTN predictive control scheme; yD-MTNC is the proposed predictive control scheme; yRBFC is the RBF model predictive control scheme; and eMTNC, eD-MTNC, and eRBFC represent the corresponding tracking errors._

We consider the control effect of the yaw velocity tracking step signal. The given reference signal is \(w_{des}=1\). The parameters of D-MTNC, MTNC, and RBFC were set as follows. The MTN predictive model adopts a 4-15-1 structure with four inputs and two powers. The initial value of the weight coefficient vector for the BP algorithm is \(\hat{w}_{\text{P}}(0)=0\) and the learning factor is \(\mu=0.5\).