Multi-Step Gaussian Regression Prediction in Dynamic Systems: A Case Study in Vehicle Lateral Control¹

Safety is one of the major concerns of the current transportation system, which has propelled research in controller design for automated vehicles (AVs). Here, every AV needs to evaluate its own control solution while avoiding the vehicles in its surrounding. AVs are generally controlled in a receding horizon fashion, which allows them to utilize information about the surrounding environment. Generally, the overall control architecture involves a high-level sampling-based trajectory planner and a low-level controller that follows the high-level waypoints [1,2]. It is imperative for any motion planning and motion control algorithm to ensure the vehicle’s safety by making sure the generated trajectories do not lead to a collision in the presence of uncertainty. To ensure safety, the decision-making algorithms should ensure that the probability with which the state distributions violate the constraints (e.g., collision avoidance) is below a given threshold. Predicting the future state probability distributions over a decision-making horizon is extremely difficult in the presence of nonlinearity in system dynamics and state- and control-dependent uncertainties, such as modeling uncertainties.

Uncertainties, including state- and control-dependent uncertainties, are common in vehicular systems. The source of these uncertainties includes uncertainty in the system parameters, state- and control-dependent noise and uncertainty, modeling error, approximation error, or other sources. These uncertainties create a mismatch between the actual system and the control-oriented model. For predictive control methods, this mismatch propagates over time and results in a significant error in long-horizon predictions. Here, we refer to the collective effect of these sources as a “mismatch” or “error.” This error can impede the operation of safety-critical systems and should be carefully quantified and included in any safety considerations. Hence, in this paper, we focus on developing an approach to model the distribution of the error over a horizon in the presence of state- and control-dependent uncertainties.

Gaussian process regression (GPR) [3,4] is a popular approach for learning the state- and control-dependent uncertainties. GPR is a flexible non-parametric model used to represent functions. It assumes that function values at any set of inputs are jointly distributed according to a multivariate normal distribution, with the mean and covariance determined by the training data. To improve computational efficiency, sparse GP approximations [5,6] have also been developed. However, these methods focus on learning the uncertainty in the current time instant or the one-step error/uncertainty, which influences the state distribution in the next time instant, but not the propagation of uncertainty over a given finite horizon. Since stochastic predictive control methods (such as stochastic model predictive control) require the error distribution over a horizon, learning one-step uncertainty is not sufficient. Even for linear systems with state- and control-dependent uncertainties, the future error distribution cannot be analytically obtained. Hence, existing methods tend to approximate it. In Ref. [7], the authors employed experimental data to find a learning-based extended kinematic model, which includes the same states as a dynamic model, and then used GP models to calculate the discrepancies between the two. However, the discrepancy is only considered in the next time-step, and the propagation of this discrepancy over time (as is pertinent to model-based controllers) is not studied. In Ref. [8], the authors modeled the entire dynamics of the vehicle with GP and incorporated the model in model predictive control (MPC). This approach disregards any information we have from first principle modeling and replaces the physics of the vehicle with black-box models. Furthermore, as the authors have mentioned, such models are not suitable for safety-constrained control, and further approximations and simplifications were done to reduce complexity and computational burden.

Some recent methods have approximated the error propagation model by linearizing system dynamics and propagating the one-step uncertainty over the horizon [3,4,9]. This can add approximation error in the propagation model. Other existing decision-making algorithms make many hard assumptions, such as linear dynamics with Gaussian uncertainty [10,11], or are overly conservative when dealing with these issues, since they consider worst-case situations and over-approximate the state distributions [12–15]. Such over-approximations reduce the solution space and decrease AV performance (e.g., increased time to reach the goal or high energy usage) and can even lead to infeasible solutions [16].

To address the research gaps, in this paper, an effort has been made to develop an approach to learn the state- and control-dependent system uncertainty and its propagation of a finite horizon. We develop a multi-stage GPR approach, where we not only learn the uncertainty in the current time instant but also its propagation over a given horizon. The main contributions of the paper include the development of a new multi-step GPR-based method to learn the uncertainty propagation model for systems with state- and control-dependent uncertainties. The contribution also includes a case study considering an AV lane change problem.

The velocity V is an external variable and can be assumed to be a time-varying function or can be obtained from a longitudinal model.

This model will be utilized later to create potential trajectories for lane changes.

To address the challenge of mismatch between the actual system and the simple model, this paper employs GPR [3,4] to provide a novel method for estimating tight bounds on the mismatch error in a prediction horizon. To train the GPR model, the hyperparameters of the model must be estimated. These hyperparameters control the smoothness of the function and the amount of noise in the data. By training the GPR model on the data from the actual system and the simple model, the mismatch error can be modeled and quantified. This provides a valuable tool for controlling the actual system with greater accuracy and safety, as the error can be accurately predicted and accounted for in the control strategy.

The method presented in this paper consists of three steps:

• Step 1 Generate a suitable set of inputs for the system.

• Step 2 Apply the input signals to the simple model and actual system, starting from the same initial condition.

• Step 3 Record the mismatch between the two as a function of input and initial condition.

• Step 4 Train the GPR models on the mismatch error.

Here, we will explain every step in detail.

To ensure accurate data collection, it is important to note that errors or mismatches are often dependent on both state and input. If a one-step prediction of the error is the sole objective, then a randomly generated set of initial conditions and input signals from an admissible set would suffice. However, for longer prediction horizons, especially in the case of predictive control, generating the necessary data becomes cumbersome. Even for a simple system with only ten input values, a ten-step prediction horizon would require 10¹⁰ input combinations, which is unfeasible to collect or train models on. Therefore, selecting input signals strategically to avoid overblowing the input domain is crucial.

Although it may appear sensible for the model to encompass every conceivable input combination, in reality, only a smaller subset of combinations is valid. For instance, in the lane change case study, gradually turning the steering to the right and then straightening it is the realistic steering combination for shifting to a lane on the right. Therefore, unrealistic input combinations that switch between left and right or high and low values should be avoided by utilizing knowledge of the system to obtain practical input combinations.

Suppose that we have explored a set of possible lane shifts, lane change times, and longitudinal speeds, resulting in n_u trajectories and corresponding feasible input sequences. However, it is important to note that when these input sequences are applied to the actual system or the simplified model, they may not result in exactly the planned trajectory. Instead, they serve as a starting point for generating potentially realistic input signals. For each input signal uⁱ(·) generated, we obtain n_t different responses from the actual system, denoted as $xji(⋅)$ to observe the effects of randomness in the system model. Let the length of this response be denoted by nⁱ. For each time instant $k=0,…,ni−H,$ we use $xji(k)$ as the initial condition and uⁱ(k : k + H) as the input signal for the simple model to obtain the response $zji(⋅∣vx,xij(k),ui(k:k+H))$⁠.

Once the error data are collected for various state and input sequences, the data are randomly split between a training and a validation set. For every prediction horizon h, we train a GPR model with the input being, at every sample, $[vxxij(k)⊤ui(k:k+h)].$ This would result in a total of H Gaussian process models for every step of the prediction, denoted as G_h(·). In practice, these models can be trained separately for every state of the system.

Figure 1 depicts the effects of the perturbations in the simple model’s response and five trajectories generated by the actual model (using the same inputs and initial conditions). The trajectories are generated for lane changes of Δy = 3.2 and Δy = 6.4 for lane changes to the right and left. Longitudinal speed is considered to belong in {15, 20, 25, 30} and t_f is 6, 7, and 8 s.

After generating the data and splitting the training and test sets, the GPR models are trained to predict the error. The results for one-step prediction are given in Fig. 2 (error in y), Fig. 3 (error in $y˙$⁠), Fig. 4 (error in Ψ), and Fig. 5 (error in $Ψ˙$⁠). Prediction results for eight-step prediction (equivalent to 4 s) are given in Fig. 6 (error in y), Fig. 7 (error in $y˙$⁠), Fig. 8 (error in Ψ), and Fig. 9 (error in $Ψ˙$⁠). It can be seen that even though for a considerably longer prediction horizon (compared to the whole lane change that can be done in 12–16 steps), we still have tight margins for our prediction. It is also worth noting that, even though the example provided here had a linear model as the starting point, the presented method does not rely on linearity or any other assumptions on the system, nor does it rely on any simplification, making it a superior method of prediction, especially for predictive-based control.

Figure 10 compares the efficacy of our method to that of Ref. [9] for predictions in y(t). The first point in the set data was used as the initial condition. Then, for the propagation method, the subsequent predictions were computed and the results were compared to the actual data. We can see that our method can maintain a high level of accuracy, while the propagation method suffers as the prediction horizon increases.

All simulations are done on a personal laptop. The computational burden of running the controller is low and can be done in real-time. The training of the GP requires more computational overhead, but as it is done offline and in advance, this does not pose a challenge to the applicability of the method. The training is done in python.

This paper presents a multi-step Gaussian process regression method for learning uncertainty propagation error in systems with state- and control-dependent uncertainties. The technique has been applied to the vehicle lateral control problem, where the error propagation model was learned during lane changes. Simulation results demonstrate the effectiveness of the proposed method in accurately predicting future state distributions and ensuring the safe operation of safety-critical systems. The presented method provides a significant improvement over existing approaches that only model current or one-step uncertainty and over-approximate the uncertainty propagation model over a horizon. The results suggest that the proposed approach has the potential to be applied to a wide range of safety-critical systems to improve their safety and performance.

This material is based upon work supported by the National Science Foundation (Grant No. $2130718$⁠).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.