From Probabilistic Pedestrian Intent to Risk-Optimal Trajectories: A Prediction-Driven Planning Framework in Shared Spaces

Abstract

With the widespread application of autonomous vehicles (AVs), their dynamic interactions with other road users pose significant challenges to trajectory planning. Previous research on trajectory planning in shared spaces has mainly focused on generating smooth trajectories, while research considering the risks of human–vehicle interactions remains insufficient. Therefore, a risk-considered trajectory planning framework for autonomous vehicles is proposed. This framework includes two modules: pedestrian trajectory prediction and vehicle planning. In the prediction module, Social-STGCNN is used to predict pedestrian trajectories, obtaining a series of trajectories and probabilities, which serve as input to the planning module. To ensure the rationality of trajectory planning, a planning model is established in Frenet coordinates based on a quintic polynomial. Combining Bayesian and equality principles, a risk-considered cost function is designed. Under this framework, the risk value is calculated using the pedestrian trajectory prediction probability, and further Bayesian and equality costs are calculated. Based on the constraints, the trajectory with the minimum cost is solved. To evaluate the rationality of this framework, we designed simulation experiments for five typical high-conflict scenarios: overtaking in the same direction, head-on collision, pedestrian crossing, encountering pedestrians from multiple directions, and turning while encountering pedestrians crossing. Simultaneously, the framework is validated in a real-world environment. The results show that the proposed method can accurately capture pedestrians’ crossing intentions and effectively avoid pedestrians. The trajectory generated in the real environment is highly consistent with that of a driver, and it exhibits excellent adaptability and robustness in high-density mixed traffic environments.

1. Introduction

With the popularization of new energy vehicles and the development of autonomous driving technology, the scenario of autonomous vehicles (AVs) sharing space with traditional road users is becoming increasingly common, spanning complex environments such as residential areas, commercial areas, and rural roads. Although current autonomous driving technology can achieve safe and reliable trajectory planning in scenarios with homogeneous traffic participants, such as highways and urban expressways [1,2,3] the complex environment of multiple traffic participants sharing roads in shared spaces poses a challenge to the trajectory planning of autonomous vehicles. The presence of diverse dynamic participants including pedestrians, electric two-wheelers, and bicycles, along with their random movement patterns and complex interactions, makes it difficult for existing technologies to generate trajectories that balance safety, efficiency, and fairness. This poses a major challenge for AV trajectory planning [4]. This study focuses on how to generate smooth and safe vehicle trajectories.

The concept of shared space, introduced in the 1970s, was defined as ‘streets or places where the dominance of motor vehicles is reduced to achieve shared space for all users, thereby enhancing pedestrian experience and comfort’ [5]. Its core characteristic is equal right-of-way, meaning that motor vehicles, non-motor vehicles, and pedestrians share the space without a clear hierarchy. This exposes autonomous vehicles to extremely high environmental uncertainties. For example, pedestrians suddenly crossing the road, electric two-wheelers randomly changing lanes, and similar behaviors can all trigger conflicts. As the cornerstone of AV safe operation, the planning module must generate feasible trajectories satisfying vehicle dynamics constraints while achieving collision-free interactions with other road users [6]. In existing research, polynomial trajectories have become the mainstream approach for trajectory description due to their excellent smoothness and trackability [7]. However, most existing studies on vehicle trajectories in shared spaces focus on generating smooth trajectories [8] and improving the right of way for vulnerable road users (VRUs) [9], lacking the allocation of risk to all road users when generating smooth trajectories. In this study, trajectory planning in shared spaces can be defined as: using the road centerline as reference line, precisely perceiving and predicting the motion trajectory of the pedestrian, and generating a local trajectory that satisfies vehicle dynamics constraints while balancing right-of-way fairness and traffic efficiency.

In summary, existing research has made some progress in trajectory planning within shared spaces, but considering the equal allocation of road usage rights to balance the risk between autonomous vehicles and pedestrians is still insufficient. To address these gaps, this paper contributes the following content:

• Designing a cost function that satisfies vehicle dynamics constraints while also considering ethical principles: This paper innovatively constructs a comprehensive cost function that integrates Bayesian principles and the principle of equality. This cost function generates vehicle trajectories that satisfy vehicle dynamic constraints while balancing safety and efficiency, reflecting fair road rights in shared spaces.
• A trajectory planning framework integrating risk components is proposed: Addressing the complexity of human–vehicle interaction and the principle of equal road rights in shared spaces, this paper introduces a trajectory planning framework incorporating risk division. By considering the variability of pedestrian trajectories and the balanced distribution of risks in shared spaces, an autonomous driving trajectory planning model is generated. This model prioritizes both driver comfort and the safety of surrounding road users.
• The safety of the framework is verified by using simulation and real scenarios: This paper is verified in five challenging simulated driving scenarios, showing excellent performance in terms of safety and efficiency. At the same time, the route planned in the real scene is highly similar to the real vehicle path, showing excellent adaptability and robustness.

The remainder of this paper is organized as follows. Section 2 reviews the related work on previous studies of deep learning and shared space. Section 3 describes the proposed trajectory planning framework, detailing the prediction module and the planning module, including the development of the comprehensive cost function. Section 4 presents the experimental validation of the proposed method, covering analysis from both simulation scenarios and real-world dataset testing. Section 5 concludes the study, summarizes the key findings, and discusses limitations and future research directions.

2. Related Work

Accurate trajectory prediction of human drivers plays a crucial role in trajectory planning for autonomous vehicles. Predicting the trajectory of human controllers is a prerequisite for achieving safe planning. Existing methods can be categorized into three types: modeling-based, learning-based, and hybrid approaches. Model-based methods model motion states through physical rules [10], yet struggle to capture the social characteristics of group interactions [11]. Among learning-based methods, approaches like Social-LSTM [12] and Social GAN [13] utilize neural networks to model interaction relationships, but exhibit insufficient accuracy in extracting spatial–temporal features. Xu et al. [14] proposed the STG-KNet model, which combines sparse graph convolutions with kernel-based structures to model evolving pedestrian interactions while maintaining structural interpretability and high computational efficiency [15]. Zhang et al. proposed a trajectory prediction model with interactive awareness and driving style awareness, which can have excellent adaptability to various driving tasks. Social-STGCNN employs spatial–temporal graph convolutional networks (STGCNN) to extract spatial–temporal features of group interactions, effectively enhancing prediction accuracy in dynamic scenarios [16]. This constitutes the core rationale for selecting this model in this paper. However, existing research mainly treats the prediction results as constraints to avoid collisions, without deeply integrating them with right-of-way allocation and risk assessment [17]. This results in planning decisions lacking an accurate response to the intentions of pedestrians.

In a trajectory planning model, the existing AV trajectory planning methods can be divided into four categories: graph search-based methods, sampling-based methods, optimization-based methods, and intent perception methods. The A*,D* and Dijkstra [18] algorithms are the most renowned methods within graph search algorithms, with various scholars having optimized and improved upon these approaches. For instance, Wang et al. [19] not only enhanced algorithmic performance but also employed a batch processing mode to simulate node expansion. This method reduces redundant work and improves cache utilization. Yuan et al. [20] proposed an improved IRRT*-D* trajectory planning algorithm for helicopter rescue operations in forest fires. This algorithm ensures that helicopters can generate safe and effective flight paths in real-time in dynamic fire environments. Common algorithms also include pure pursuit [21,22] algorithms and the Stanley controller [23]. Sampling-based methods, such as RRT-Connect generate trajectories by randomly sampling and connecting state points [24,25], but have low computational efficiency and cannot meet real-time requirements. Numerical optimization methods solve for the optimal trajectory by constructing an objective function, but constraint design often ignores fair right-of-way allocation. Intention-aware methods attempt to make decisions by integrating environmental interaction information, but are not accurate enough in recognizing human control intentions [26]. In response to the above problems, Liu et al. [27] proposed the Hybrid Predictive Integrated Planning (HPP) framework, which solves the problem of inconsistency between future prediction and planning interaction, but does not include vehicle dynamics constraints in planning. Yan et al. [28] proposed a safety-oriented hierarchical personalized driving system to alleviate preference conflicts between passengers and the intelligent vehicle control system. Zheng et al. [29] applied the diffusion model to end-to-end closed-loop autonomous driving planning. By utilizing the inherent classifier guidance mechanism of the model, the driving behavior is dynamically adjusted during the reasoning process without the need for retraining. However, the end-to-end autonomous trajectory planning model has weak interpretability and performs poorly when there are many vehicles moving laterally.

In a shared space scenario, some scholars employ Frenet coordinate systems to achieve longitudinal–lateral trajectory decoupling planning, enhancing trajectory smoothness [6], yet failing to incorporate human-controller right-of-way demands as constraints. Geisslinger et al. [30] propose an ethics-based trajectory evaluation method, but it only takes into account the risks of different road users, ignoring vehicle motion planning. Huang et al. [31] employed game theory to resolve conflicts, yet neglected the impact of vehicle dynamics constraints on trajectory feasibility. Guo et al. [32] developed an injury prediction model for anticipated scenarios addressing the high accident fatality and injury rates among electric two-wheeled vehicles. They proposed a trajectory planning framework for AVs based on head injury prediction for VRUs, but paid little attention to the smoothness of vehicle trajectories. In summary, at the current stage, vehicle trajectory planning in shared spaces lacks research that simultaneously considers both vehicle motion planning and the equal allocation of road rights characteristic of shared spaces, specifically the equitable distribution of road usage risks.

The inherent equal right-of-way in shared spaces presents new challenges for the trajectory generation of AVs: a balance must be struck between fairness and efficiency in risk allocation [33]. Some researchers have enhanced interaction transparency through external human–machine interfaces [34], yet failed to address road rights at the algorithmic level. Li et al. [35] proposed a dynamic road rights allocation method, but did not integrate trajectory prediction with vehicle dynamics constraints. Wang et al. [36] proposed a two-stage trajectory planning method for AVs in socially uncertain intersection scenarios that accounts for other road users, evaluating each candidate planning trajectory through a safety-balanced value function.

Regarding risk assessment, Harsanyi’s Bayesian principle [37] solely pursues overall risk minimization. Building upon this, Geisslinger et al. [30] proposed a more comprehensive ethical framework that combines Bayesian principles with equality principles to create an ethical cost function. However, neither approach incorporates vehicle dynamics constraints.

3. Methodology

The trajectory planning is formulated as an optimization problem considering the risk of conflicts between AVs and human-controllers in spared spaces, as shown in Figure 1. The system consists of a prediction module and a planning module. After extracting road information, the human controller trajectory is fed into the prediction module to obtain a series of predicted trajectories and their corresponding probabilities. Simultaneously, the vehicle’s trajectory is decomposed into lateral and longitudinal planning in the Frenet coordinate system. Within this planning framework, the predicted human controller trajectory serves as input for calculating the risk cost. Subsequently, trajectories are filtered, and the one that satisfies constraints and minimizes cost is selected as the final trajectory output.

3.1. Prediction Module

Pedestrians’ motion prediction is a typical sequential generation problem. Numerous mature trajectory prediction methods already exist, such as Social-LSTM [38] and TUTR [39]. In order to better capture pedestrians’ interactions and social influences within the environment, this paper employs Social-STGCNN [16] to predict the trajectories of pedestrians. The pedestrians prediction problem can be defined as: Given a group of $N$ pedestrians with their matching observed trajectories $t{r}_o^n$ in a scenario, $n\in \{1,\dots ,N\}$ over a time period $T_o$, the future trajectories $t{r}_p^n$ need to be predicted over a future time horizon $T_p$. For a pedestrian $n$, the matching trajectory is written to be predicted as $t{r}_p^n=\{\left.{\mathbf{P}}_t^n=({\mathbf{X}}_t^n,{\mathbf{Y}}_t^n)\right|\mathrm{t}\in \{1,\dots {,\mathrm{T}}_p\left\}\right\}$, where $({\mathbf{X}}_t^n,{\mathbf{Y}}_t^n)$ are random variables describing the probability distribution of the location of pedestrians $n$ at time $t$, in the 2D space. It is assumed that $({\mathbf{X}}_t^n,{\mathbf{Y}}_t^n)$ follows a bi-variate Gaussian distribution such that ${\mathbf{P}}_t^n~\mathcal{N}({\mu }_t^n,{\sigma }_t^n,{\rho }_t^n)$. In addition, the predicted position is denoted as ${\widehat{\mathbf{P}}}_t^n$, which follows the estimated bi-variate distribution f. This estimate is based on the pedestrians’ historical position at time t, as well as the historical trajectories of its adjacent road users. Social-STGCNN [16] is trained to minimize the negative log-likelihood which is defined as:

$$Z^n(\mathbf{W})=-{\displaystyle \sum _{t=1}^{{\mathrm{T}}_p}\log (\mathbb{P}(({\mathbf{P}}_t^n|{\widehat{\mu }}_t^n,{\widehat{\sigma }}_t^n,{\widehat{\rho }}_t^n)})$$

(1)

where $\mathbf{W}$ includes all the trainable parameters of the model, ${\widehat{\mu }}_t^n$ is the mean of the distribution, ${\widehat{\sigma }}_t^n$ is the variances, and ${\widehat{\rho }}_t^n$ is the correlation.

The performance of the trained trajectories is evaluated using the Average Displacement Error (ADE) [40] and Final Displacement Error (FDE) [38], where ADE is defined by Equation (2) and FDE is defined by Equation (3). ADE evaluates the average distance difference between the predicted position and the true position at each time point, while FDE assesses the distance difference between the final predicted position and the true position. Since Social-STGCNN generates a bivariate Gaussian distribution as the prediction result, to compare the distribution with the target values, we follow the evaluation method adopted by Social-LSTM: generate 20 samples based on the predicted distribution, then select the sample closest to the true value to compute ADE and FDE.

$$ADE={\displaystyle \frac{{\displaystyle \sum _{n\in N}{\displaystyle \sum _{t\in T_p}{‖{\widehat{\mathbf{P}}}_t^n-{\mathbf{P}}_t^n‖}_2}}}{N\times T_p}}$$

(2)

$$FDE={\displaystyle \frac{{\displaystyle \sum _{n\in N}{‖{\widehat{\mathbf{P}}}_t^n-{\mathbf{P}}_t^n‖}_2}}{N}},t=T_p$$

(3)

The final output format of the Social-STGCNN is as per Equations (4) and (5), which is then fed into the planning module.

$$M=\left[\begin{array}{c}\left[m_1^{(t+1)},m_1^{(t+2)},\dots ,m_1^{(t+T_p)}\right]\\ \left[m_2^{(t+1)},m_2^{(t+2)},\dots ,m_2^{(t+T_p)}\right]\\ ⋮\\ \left[m_N^{(t+1)},m_N^{(t+2)},\dots ,m_N^{(t+T_p)}\right]\end{array}\right]$$

(4)

$$m_n^{(t+k)}=\left[{\{(x_n^{t+k},y_n^{t+k})\}}_{s=1}^S,{\widehat{\mu }}_{x,n,t+k},{\widehat{\mu }}_{y,n,t+k},{\widehat{\sigma }}_{x,n,t+k},{\widehat{\sigma }}_{y,n,t+k},{\widehat{\rho }}_{n,t+k}\right]$$

(5)

where $M_i$ is a matrix of $N\times T_p$, representing the predicted trajectory of the $N$ pedestrian in the $i$ frame; $T_p$ denotes the number of time steps to be predicted; $m_n^{(t+k)}$ represents the predicted state of the $n$ pedestrian at time $t+k$, ${\{(x_n^{t+k},y_n^{t+k})\}}_{s=1}^S$ denotes the set of $S$ points ($S=20$) sampled from the predicted bi-variate Gaussian distribution.

3.2. Planning Module

To ensure the continuity and smoothness of trajectories computed across different time intervals, this paper will perform longitudinal and lateral decoupling of the desired trajectories in Frenet coordinate.

3.2.1. Frenet Coordinate

The Frenet coordinate system first requires the establishment of a reference line. In this study, the reference line is defined as the centerline (grey-line) of the road as illustrated in Figure 2.

Frenet coordinates was defined as $(s,l)$, where $s$ denotes the distance from the reference line’s origin to the corresponding point on the reference line matching the center of mass $A(x_A,y_A)$ of the object under study. The calculation is as follows:

$$l=\left\{\begin{array}{c}{‖l‖}_2=\sqrt{{(x_A-x_O)}^2+{(y_A-y_O)}^2}\\ sign(l)=\mathit{T}\times \mathit{A}\end{array}\right.$$

(6)

where $\mathit{T}$ denotes the unit tangent vector passing through point $O$, $\mathit{A}$ denotes the vector with origin at $O$ and endpoint at $A$, and ${‖\cdot ‖}_2$ denotes the L2 norm. The sign of $l$ is determined by the sign of $\mathit{T}\times \mathit{A}$. To simplify calculations, the vector $\mathit{A}$ in the text is taken as the unit normal vector passing through point $O$, i.e., $N$ in Figure 3.

Therefore, in the Frenet coordinate, the correspondence between the matching point $\mathit{r}(s)$ and the point of interest $\mathit{x}(s,l)$ is as follows:

$$\mathit{x}(s(t),l(t))=\mathit{r}(s(t))+l(t){\mathit{N}}_r(s(t))$$

(7)

Reference Line Description. The reference line $s(t)$ denotes the cumulative arc length at time $t$, defined as

$$s(t)={\displaystyle {\int }_0^t{‖r^{\prime }(x)‖}_{2}dx}$$

(8)

where $r^{\prime }(t)={\displaystyle \frac{\partial r}{\partial t}}$, ${‖\cdot ‖}_2$ denote the L2 norm. This paper defines the reference line as the centerline of a structured road. To compute the cumulative arc length of the reference line, it must be discretized. From Equation (8) $s(t)\equiv s\Rightarrow t=t(s),s(t)\equiv s$, it is evident that $s$ is not only a function of $t$, but $t$ is also a function of the cumulative arc length $s$. Therefore, $r(t)$ may be expressed as $r(s)$.

Selection of matching points. In practical scenarios, reference lines require discretization $v_j(j=0,1,\dots ,i-1,i,i+1,\dots ,n)$. However, differing discretization accuracies may result in the vehicle with the smallest Euclidean distance after discretization not being the optimal matching point $v_j$ during selection. Consequently, matching point indices must be calculated and selected. The selection process is described as follows:

(1) Calculate $v_{index}=\arg \min l_j,(j=0,1,\dots ,n)$ with the smallest Euclidean distance from point $A(x_A,y_A)$. Using this point’s index, locate the indices of the preceding and subsequent points, as illustrated in Figure 4. In the diagram, the index of the point with the smallest distance is $v_i$, while the indices of the preceding and subsequent points are $v_{i-1}$ and $v_{i+1}$, respectively.

(2) Based on the previous index $v_{i-1}$ of the minimum point and the vehicle’s current trajectory point $A(x_A,y_A)$, calculate the projection $A_r$ of ${\mathit{v}}_{i-1}\mathit{A}$ onto ${\mathit{v}}_{i-1}{\mathit{v}}_{i+1}$; this constitutes the matched point.

3.2.2. Longitudinal and Lateral Trajectory Planning

Following decoupling via Frenet coordinates, an AV’s path planning problem can be divided into lateral motion planning and longitudinal motion planning. Unlike the pure pursuit algorithm [21], in the Frenet coordinate system, longitudinal motion planning primarily concerns vehicle velocity planning, whilst lateral motion planning focuses on steering planning. Longitudinal motion planning employs different solution methods depending on obstacle presence, with common approaches being time-based fourth-order polynomial and time-based fifth-order polynomial methods [41]. Solutions for lateral motion planning primarily employ time-based quintic polynomial and quadratic programming methods. To conserve computational resources, this paper employs time-based quintic polynomials for both longitudinal and lateral motion planning. Longitudinal motion is denoted as $s(t)$, lateral motion as $l(t)$. Recognizing their identical functional structure, a generic quintic polynomial function $m(t)$ is introduced for unified expression:

$$o(t)=h_0+h_{1}t+h_2{t}^2+h_3{t}^3+h_4{t}^4+h_5{t}^5$$

(9)

where $h_i\in ℝ,i=0,1,\dots ,5$ denotes the polynomial coefficient. Specifically, the longitudinal trajectory $s(t)$ corresponds to coefficient $\{a_0,a_1,\dots ,a_5\}$, whilst the transverse trajectory $l(t)$ corresponds to coefficient $\{b_0,b_1,\dots ,b_5\}$.

Given the initial state $O_0$, the final state $O_T$ of the AV is sampled at discrete time points in both the longitudinal and lateral directions. Subsequently, the initial and final states are substituted into Equation (10) to solve for the coefficients of the trajectory polynomial $o(t)$.

$$O_0=[o(0),\dot{o}(0),\ddot{o}(0)]$$

(10)

$$O_T=[o(T),\dot{o}(T),\ddot{o}(T)]$$

(11)

where $O_i,i=0,1,\dots ,T$ represents the longitudinal trajectory $s(t)$ and the transverse trajectory $l(t)$. It follows that:

$$h_0=o(0)$$

(12)

$$h_1=\dot{o}(0)$$

(13)

$$h_2={\displaystyle \frac{\ddot{o}(0)}{2}}$$

(14)

$$\left[\begin{array}{ccc}{T}^3& T^4& T^5\\ 3T^2& 4T^3& 5T^4\\ 6T& 12T^2& 20T^3\end{array}\right]\times \left[\begin{array}{c}{h}_3\\ h_4\\ h_5\end{array}\right]=\left[\begin{array}{c}o\left(T\right)-[(o\left(0\right)+\dot{o}(0)T+\ddot{o}(0)T^2)/2]\\ \dot{o}\left(T\right)-(\dot{o}\left(0\right)+\ddot{o}\left(0\right)T)\\ \ddot{o}\left(T\right)-\ddot{o}\left(0\right)\end{array}\right]$$

(15)

3.2.3. Longitudinal and Lateral Trajectory Assessment

Having obtained multiple longitudinal and lateral trajectories corresponding to different time points, it is necessary to evaluate these trajectories and select the most suitable one as the AV’s trajectory. For the scenario under investigation in this paper, trajectory evaluation considers two aspects: the smoothness of the vehicle’s route and the risk borne by surrounding road users.

Trajectory smoothing cost. The cost of route smoothness for vehicles, denoted as $C_{traj}$, is derived by summing the $C_d$ and $C_s$.

$$C_{traj}=C_s+C_d$$

(16)

where the smoothing cost of the AV’s trajectory in both longitudinal and lateral directions is given by Equation (17)

$$C_o={\omega }_{o1}{\displaystyle {\int }_0^T{\stackrel{⃛}{o}}^2(t)}dt+{\omega }_{o2}{\displaystyle {\int }_0^T{\ddot{o}}^2(t)}dt+{\omega }_{\mathrm{o}3}f{(o(t))}^2$$

(17)

where $C_o$ represents $C_s$ and $C_d$, the jerk term smooths the trajectory, the acceleration term suppresses abrupt velocity changes, and the terminal state term aligns the AV’s terminal state with the desired state.

$$f(o(t))=\left\{\begin{array}{l}d(T),lateral\hfill \\ s(T)-s_{target},longitudinal(stopping)\hfill \\ \dot{s}(T)-{\dot{s}}_{target}longitudinal(velocitykeeping)\hfill \end{array}\right.$$

(18)

where $d(T)$, $s(T)$, and $\dot{s}(T)$ represent the position and velocity of the lateral and longitudinal terminal states, respectively; $s_{target}$ and ${\dot{s}}_{target}$ denote the target position and velocity of the AV itself. In practice, whether an autonomous vehicle chooses to stop or follow is determined by the cost function’s ranking. Should the cost of stopping be lower than that of following, the autonomous vehicle will opt to halt. Similarly, should the cost of changing lanes be lower than that of following, the vehicle will select lane-changing. This demonstrates that the autonomous vehicle’s process of selecting the minimum cost constitutes its decision-making process [8].

Trajectory risk cost. In shared spaces, AVs travel at slower speeds due to the complex environment, resulting in relatively minor injuries when collisions occur compared to other environments. Furthermore, the allocation of right-of-way between pedestrians and vehicles is relatively equitable. It is biased to consider only the risks of AVs themselves while ignoring the safety of human-controllers, or to focus only on the risks of AVs themselves while ignoring the risks of human-controllers. As noted by the European Commission [42] and prior research [43], a singular risk allocation principle fails to meet ethical decision-making requirements. To reasonably distribute and evaluate risks during interactions between the AVs and other road users, while accounting for equitable right-of-way allocation within shared spaces, this paper introduces a risk value metric. This approach considers the risks faced by multiple interacting parties when assessing the AV’s trajectory. Drawing upon the injury values outlined in reference [30], the paper calculates the potential harm that other road users in the vicinity of the AVs may sustain.

$$R_{traj}(o)=\max (P(o)H(o))$$

(19)

where $P(o)$ represents the probability of trajectory $o$ for that human-controller. It should be noted that the primary subject of this study was AVs. As previously mentioned, an AV’s path is obtained through point scattering. The risk value is calculated only when the AV’s planned path overlaps with that of other road users. The specific calculation of $H(o)$ is given by Equations (20) and (21).

$$\Delta v_A={\displaystyle \frac{m_B}{m_A+m_B}}\sqrt{{v_A}^2+{v_B}^2-2v_A{v}_B\cos \alpha }$$

(20)

$$H={\displaystyle \frac{1}{1+e^{c_0-c_1\Delta v-c_{area}}}}$$

(21)

where $m$ and $v$ represent the mass and velocity of two road users, and $\alpha$ denotes the collision angle; $c_0$, $c_1$, and $c_{area}$ are empirical coefficients [30]. The probability of accidents with severity MAIS3+ [44] serves as a comparative benchmark, mapping injuries to values between 0 and 1.

After calculating the risks for all road users, the trajectory with the lowest overall risk is selected as the final output trajectory through risk classification. The specific principles for risk classification and the calculation of risk costs will be detailed subsequently.

Principles of Risk Classification: Due to the inherent uncertainty in predicting the trajectories of other road users during an AV’s path planning, this paper calculates and allocates risks arising from such uncertainty. Different risk appetites lead to different decision-making processes. To balance the risks faced by autonomous vehicles among other road users, this paper introduces Bayesian principles and the principle of equality to evaluate the path taken by autonomous vehicles. This ensures that, under shared space conditions, different road users face risks and are treated fairly, enabling autonomous vehicles to make rational decisions.

The significance of Bayesian decision principles [37] in ethics was articulated as early as 1978, noting that Bayesian principles adhere to a utilitarian approach—namely, minimizing overall risk. Within this paper’s scenario, this entails minimizing the aggregate risk for multiple parties including AVs and vulnerable road users. This approach satisfies risk preferences and societal acceptability within shared space scenarios.

$$C_B(o)={\displaystyle \frac{{\displaystyle \sum _{i=1}^{\left|S_R\right|}{R}_i(o)}}{\left|S_R\right|}}$$

(22)

where $C_B(o)$ denotes the Bayesian cost of trajectory $o$, $\left|S_R\right|$ represents the set of all road users, and $R_i(o)$ signifies the risk incurred by the $i$ road user under trajectory $o$.

However, while Bayes’ principle can minimize overall risk, it ignores the risk borne by each entity, potentially leading to one party bearing excessive risk and failing to consider the fairness of risk allocation. For example, as shown in Figure 5, if the total risk of situation 1 is 0.6 and the total risk of situation 2 is 0.55, Bayes’ principle would tend to choose the path of situation 2. However, this exposes another pedestrian to extremely high risk, failing to consider the fairness of risk allocation.

To mitigate such occurrences, invoking the principle of equality cited in Reference [30] ensures that the application of Bayesian principles does not introduce biases favoring abnormally high risks for individual road users relative to low overall risks. The principle of equality effectively addresses the fragile inequality problem among human-controllers. Equation (23) quantifies and constrains the differences in risk exposure among different human-controllers, thereby maintaining fairness in risk allocation while pursuing the minimization of overall risk.

$$C_E(o)={\displaystyle \frac{{\displaystyle \sum _{i=1}^{\left|S_R\right|}{\displaystyle \sum _{j=1}^{\left|S_R\right|}\left|R_i(o)-R_j(o)\right|}}}{{\displaystyle \sum _{k=1}^{\left|S_R\right|-1}k}}}$$

(23)

where $C_E(o)$ denotes the cost of the equality principle for trajectory $o$, $\left|S_R\right|$ represents the set of road users, while $R_i(o)$ and $R_j(o)$ denote the risks for the $i$ and $j$ road users, respectively, under trajectory $o$.

The denominator undergoes normalization to render the cost of the equality principle independent of the number of road users. The introduction of the Bayesian principle and the equality principle embody the equitable distribution of risks within shared spaces.

$$C_{risk}(o)={\omega }_B{C}_B(o)+{\omega }_E{C}_E(o)$$

(24)

The cost of evaluating the longitudinal and transverse trajectories is given by Equation (24). The optimal combined trajectory $o(t)$ is ultimately obtained by solving the following optimization problem:

$$\arg \underset{o}{\min }C(o)=C_{traj}+C_{risk}(o)$$

(25)

$$\begin{array}{l}s.t.\\ \left|k(t)\right|\le k_{\max }\\ \left|a(t)\right|\le a_{\max }\\ R(o)\le R_{\max }\\ d(o(t),B_i)\ge d_{min}\end{array}$$

(26)

where $k(t)$ denotes curvature; considering trajectory smoothness, $k_{max}$ represents the maximum curvature of the trajectory; $a(t)$ denotes acceleration; $a_{max}$ denotes maximum acceleration; $d$ denotes the distance between the vehicle and the obstacle; $d_{min}$ denotes the minimum safe distance; and $B_i$ denotes the road boundary. By solving the equations, the trajectory with the lowest cost under the constraints is obtained and output as the AV trajectory.

4. Experiment

To assess the feasibility of the planning scheme, this paper simulates five complex scenarios based on documents issued by the European Union [44], and conducts testing on the Ind dataset [45].

4.1. Simulation Experiment

Five distinct scenarios as shown in Figure 6 were designed to test this method: a. overtaking in the same direction, b. head-on collision, c. pedestrian crossing collision, d. collision with multiple pedestrians travelling in the same and opposite directions, and e. encountering a crossing pedestrian while turning. The vehicle commenced movement from a stationary position.

4.1.1. Overtaking in the Same Direction

Scenario a: The vehicle departs from the left edge of the screen. Three pedestrians set off in the same direction from coordinates (12, 5), (5, 10), and (8, 3). The vehicle’s endpoint is set at (20, 9). Pedestrian 0 moves at a constant velocity of 1.5 m/s along the y-axis, oscillating in the x-axis with no net displacement. Pedestrian 1 moves uniformly along the x-axis at 1.5 m/s, oscillating along the y-axis with no net displacement. Pedestrian 2 moves diagonally at 2.33 m/s.

Pedestrian 0’s trajectory is set as follows:

$$\left\{\begin{array}{l}x=12+sin\left(t\right)\\ y=5+1.5t\end{array}\right.$$

(27)

Pedestrian 1’s trajectory is set as follows:

$$\left\{\begin{array}{l} x=5+1.5t\\ y=10+2sin\left(0.3t\right)\end{array}\right.$$

(28)

Pedestrian 2’s trajectory is set as follows:

$$\left\{\begin{array}{l} x=8+2t+1.5sin\left(0.4t\right)\\ y=3+1.2t+1.2sin\left(0.6t\right)\end{array}\right.$$

(29)

As shown in Figure 7, the AV detects a pedestrian walking in the same direction ahead at time step 3 and opts to steer slightly to the right at this moment. However, at the subsequent time step t = 4, it chooses to maintain straight-line travel. Upon reaching the point of interaction with the pedestrian at the fifth time step, it achieves a maximum speed of 15.9 km/h before turning left and gradually commencing deceleration to slow down.

The speed variation graph, Figure 8a, shows that the AV gradually accelerates from step 27 and continues its journey. The cost of the autonomous vehicle also fluctuates as it approaches the pedestrian. As shown in Figure 8b, the total cost of the AV increases sharply to 2908.67331 in step 5. In the simulated scenario, the AV chooses to decelerate and then begins to turn left.

4.1.2. Head-On Collision

Scenario b: The vehicle departs from the left edge of the screen (2, 4.5). Two pedestrians set off from coordinates (15, 5) and (15, 6), respectively, moving in the opposite direction to the vehicle. The vehicle’s destination is set at (20, 10). Both pedestrians travel along the x-axis at a speed of 1.5 m/s. This scenario simulates an encounter between the vehicle and two pedestrians walking side by side.

As can be seen from Figure 9 the vehicle detected the pedestrian ahead at time step 3, opted for a slight steering manoeuvre to avoid the pedestrian, and completed the evasive action by time step 10. Throughout the entire process, the vehicle did not decelerate but proceeded at a constant speed through the junction.

At each time step, the path selected by the AV, as shown in Figure 10, exhibits its highest total cost at time step 3, reaching approximately 50. At this point, the AV opts to turn slightly to the right. Throughout the entire simulation, only time steps 2, 3, and 4 produced relatively low risk costs. This indicates that during the simulation, the relative distances between the AV and the two simulated pedestrians remained substantial, with virtually no conflicts occurring.

4.1.3. Pedestrian Crossing Collision

Scenario c: A vehicle is traveling from north to south while pedestrians cross the street. The vehicle’s starting point is (12, 6), and the pedestrians’ starting points are (15, 5) and (15, 6). The pedestrians cross the street at a speed of 1 m/s, moving along the direction perpendicular to the reference line. The vehicle meets the pedestrians at step 12.

As shown in Figure 11b, at time step 7, the vehicle detected the pedestrian and initiated deceleration. By time step 8, the speed had decreased from 15.2 km/h to 15.0 km/h, commencing a gradual reduction. Upon reaching the pedestrian crossing point at 12.4 km/h, the vehicle subsequently accelerated towards its target speed.

Throughout the entire process, the path chosen by the AV did not incur any risk costs, as shown in Figure 12, meaning that at each time step, the selected trajectory remained relatively distant from pedestrians. Furthermore, the behavior of gradually decelerating upon detecting pedestrians aligns with ethical constraints.

4.1.4. Encountering Pedestrians from Multiple Directions

Scenario d: The vehicle starts at (2, 8.5) and travels along an L-shaped road to (30, 15). Eight pedestrians walk in the opposite direction to the AV at a constant speed of 1.5 m/s. Throughout the process, the pedestrians mainly move in a straight line. The encounter distance between the AV and the pedestrians is set long enough to ensure that the AV can fully accelerate to its target speed. This is to explore whether the AV can react quickly to pedestrians encountered ahead while traveling at high speed. As shown in Figure 13, the AV accelerates to approximately 20 km/h in step 15, detects pedestrians walking in the opposite direction ahead, and then begins to decelerate. Five steps later, in step 20, it decelerates to 18 km/h, then decelerates again, coming to a complete stop in step 30. In step 26, it turns slightly left, moves towards the center, and moves a little further away from the pedestrians. Unlike Scenario b, the AV adopts a conservative driving approach in this simulation, rather than an aggressive one. This meets the requirements for ensuring the safety of both AV and human-controlled vehicles in a shared space.

At time step 31, the AV commences driving once more, gradually accelerating to its initial target speed. As illustrated in Figure 14, within complex scenarios, when vehicle speeds are low, risk costs constitute a significant proportion of the total cost, exhibiting a consistent trend.

4.1.5. Encountering Pedestrians Crossing While Turning

Scenario e: The vehicle starts at (−5, 4.5) and travels along an L-shaped road towards (12, 10). Pedestrians come from (15, 5) and (9.5, 6), respectively, crossing the turn in opposite directions. The autonomous vehicle must yield to both pedestrians at the turn before reaching its destination. Both pedestrians are traveling at a speed of 1.3 m/s. The encounter distance between the AV and the pedestrians is also set long enough to ensure that the AV can fully accelerate to its target speed.

As shown in Figure 15 and Figure 16, the AV accelerates to 15.1 km/h in step 8 and continues to accelerate. In step 12, it detects a pedestrian, begins to decelerate from 16.8 km/h, and finally comes to a complete stop in step 23. In step 15, it begins to turn left, but is disturbed by a pedestrian, so it slightly turns right to avoid it.

4.2. Real-World Experiment

To validate the practicality of the proposed method, verification was conducted using the inD [45] dataset collected from real-world scenarios. The inD dataset provides several scenarios depicting the movement of pedestrians and vehicles within shared spaces. Vehicles are operated by human drivers, with all trajectories of both vehicles and pedestrians recorded. This study selected a scenario involving one vehicle and two pedestrians. The vehicle would turn through the intersection, while the pedestrians would move from top to bottom across the crossing. The trajectories extracted for the pedestrians and the vehicle driven by a human driver are shown in Figure 17.

During the experiment, the vehicle is assumed to be the AV. Therefore, only the starting point, destination point, and initial velocity of the vehicle need to be set to match the real-world scenario, whilst the vehicle’s trajectory is planned using the proposed method.

To validate the performance of the pedestrian prediction module, Social-STGCNN was employed to forecast pedestrian movement trajectories. A total of 14,100 pedestrian trajectories were extracted, with 9870 data points utilized for model training, 1410 for testing, and 2820 for validation. The hyperparameter settings are detailed in

Table 1. Hyperparameter configuration.

Parameter
Batch size	128
Epoch	250
STGCNN	1
TXPCNN	5
Learning Rate	0.01

.

Following training, the ADE was 0.25 and the FDE was 0.21. To validate the model’s performance, training was conducted using Social-LSTM for comparison. On Social-LSTM, the ADE was 0.33 and the FDE was 0.28, demonstrating that Social-STGCNN overall performed better than Social-LSTM. The pedestrian trajectory prediction results are shown in Figure 18.

During testing, the trajectory with the highest probability is employed as the pedestrian’s trajectory for the next moment. The remaining possible trajectories at the same moment are also incorporated into the cost calculation for the vehicle’s trajectory. It can be observed that the proposed method achieves high accuracy in predicting pedestrian movement trajectories. The vehicle trajectory is shown in Figure 19, with boxes marking the movement paths of both the pedestrian and vehicle for the preceding 13 s. It can be observed that the vehicle accurately anticipates the pedestrian’s movement and successfully identifies suitable space to navigate through the crowd without any collisions.

As shown in Figure 20a, the simulated vehicle begins decelerating at time step 9 and slows to 9.5 km/h by time step 15, passing through the intersection at a reduced speed. The cost changes for the simulated vehicle are depicted in Figure 20b, revealing that as deceleration occurs at time step 15, the vehicle interacts with pedestrians, causing the risk cost to start increasing.

Comparing the actual vehicle trajectory with the simulated vehicle trajectory, Figure 20, it can be observed that the simulated vehicle planning trajectory is smoother than the actual trajectory. Furthermore, the generated trajectory of the simulated vehicle is largely consistent with that of a human driver.

5. Conclusions

This paper proposes a vehicle trajectory planning model incorporating risk division to address the complex vehicle trajectory planning in shared spaces involving human–vehicle interactions. In the trajectory prediction module, the Social-STGCNN model is used to predict the future trajectory and probability of the human-controller. This effectively captures the spatiotemporal interaction features between the human and controller, and its prediction accuracy is validated on the InD dataset: ADE = 0.25, FDE = 0.21, significantly outperforming the traditional Social-LSTM method. In the trajectory planning module, the Frenet coordinate system is adopted, and a fifth-order polynomial is used to decouple longitudinal and lateral motion planning. A comprehensive cost function is constructed, innovatively combining Bayesian principles with the principle of equality, thereby achieving reasonable risk allocation in complex dynamic interaction scenarios. This method satisfies the constraints of vehicle dynamics while generating safe, smooth, and ethical trajectories in complex environments. Simulated and real-world dataset validations demonstrate that the proposed planning system can accurately predict the human-controller’s motion intentions and proactively adjust the vehicle’s trajectory. Specifically, the five scenarios demonstrate that within this framework, AVs and pedestrians can share the right of way in shared spaces, allowing efficient passage while also taking into account that pedestrians, as VRUs, are vulnerable. For example, in scenario e, the AV only slows down by 1 km/h, passing at 14 km/h rather than stopping directly to yield; scenarios a and d show that in complex environments, autonomous vehicles choose a conservative driving mode, slowing down to yield without coming to a complete stop. This method avoids collisions in shared spaces while balancing traffic efficiency and fair right-of-way allocation. Especially in high-density mixed traffic environments, the system exhibits conservatism and robustness, and the generated trajectories are highly consistent with the behavior of human drivers.

This study has several limitations. 1. The simulation environment represents a simplified version of shared spaces; future research will test the framework in more chaotic, high-density crowds. 2. Although the framework focuses on pedestrians, future versions will introduce diverse agents, such as cyclists. 3. The current collision injury calculation uses fixed parameters, which may not accurately assess the injuries suffered by different types of road users. In the future, efforts will be made to develop a more adaptive risk cost mechanism to better evaluate the injuries of different types of road users.