Various cooperative control architectures have been proposed in (Saleh et al., 2013; Soualmi et al., 2014; Nguyen et al., 2017; Schnelle et al., 2017; Wang et al., 2017) based on DiL designs. In (Abbink et al., 2011; Mars et al., 2014; Boink et al., 2014) haptic feedback from steering wheel was used to ensure both driver and the autonomous controller participated in the driving action. In (Wada et al., 2016), a co-operative control approach for lane keeping based on H ₂ preview control was proposed by incorporating a nuero-muscular driver model. Similarly in (Soualmi et al., 2014), a haptic shared control between driver and e-copilot considered the use of driver torque as haptic feedback to design T-S fuzzy controllers for lane keeping. In (Wang et al., 2017), for varying driver steering characteristics such as delays, and preview time, a DiL gain-scheduling H _∞ robust shared controller was proposed. In, (Wada et al., 2016; Saito et al., 2018) based on cooperative status detection, a conflict free smooth transition of authority between human driver and autonomous controller was proposed. Similarly in (Oudainia et al., 2022), conflict mitigation by adapting the cost function objective was proposed. Extending the work of (Nguyen et al., 2015), a co-operative control approach employing T-S models was proposed in (Nguyen et al., 2017) to perform lane keeping and conflict minimization simultaneously. In (Lv et al., 2018) a haptic control architecture was developed for smooth transition of control authority with adaptation to driver cognitive workload. The works in (Wada et al., 2016; Wang et al., 2017) assumed constant longitudinal speed in the design of lane keeping controllers. Further, conflicts between driver and the automated driving system was not explicitly addressed in (Oufroukh and Mammar, 2014; Wang et al., 2017). In (Nguyen et al., 2018), the unpredictable driver-automation interaction is explicitly taken into account in the shared control scheme, via a fictive driver activity variable. Similarly, (Wang et al., 2018) proposed a shared controller for path tracking considering different drivers’ characteristics to reduce the physical workloads of the inexperienced drivers. The concept of driver-automation oriented for shared control of lane keeping assist (LKA) was proposed in (Sentouh et al., 2018) based on two local optimal controllers combining system perception with robust control so that the proposed strategy can successfully share the control authority between human drivers and the LKA system. (Pano et al., 2020) presented a shared control strategy with a feedforward-feedback synthesis using a mixed H ₂/H _∞ control involving both lane following performance and sharing capabilities indicators. This strategy is interesting as it used the sharing level between the human driver and the assistance explicitly. Moreover, the shared control proposed in (Zhao et al., 2020) is much more flexible since this work follow-up the study in (Pano et al., 2019). It aims to integrate the driver’s adaptation in the level of sharing over the driving task using cybernetic driver model. The control authority transition from automated functionality to a human driver is particularly studied in (Lv et al., 2021; Oudainia et al., 2022). This helps drivers to take control of the vehicle progressively. Recently, the shared control strategy have been studied considering driver’s behaviors to get a better cooperation. (Huang et al., 2019) presents a cooperative framework developed by adopting data-driven adaptive dynamic programming and an iterative learning scheme based on classical small-gain theory. In (Chen et al., 2020), the lane-keeping control method adopts a driver model based on near and far visual angles with a robust parallel distributed compensation H _∞ controller. These approaches typically validated the cooperative performance of the DiL design for lane keeping tasks. Meanwhile, it is necessary to ensure that this new type of mobility takes into account both driver expectations and foreseeable changes in road user behavior (Sentouh et al., 2009). Overall, the shared control issue being able to share the driving responsibility with human drivers still remains challenging. It is therefore necessary to develop a new Human Machine Interaction strategy (HMI) that allows gradual shared control of vehicle commands between the vehicle and the driver.

Within this work, a novel shared control architecture shown in Figure 1, based on the dynamic of the driver’s visual and neruo-muscular parameters which are identified online will be developed and validated in real-time under various driving scenarios with the SHERPA-LAMIH driving simulator. This strategy considers the driver’s visual and neuro-muscular properties provide an adequate assistance for the driver’s abilities. Based on the principle of shared control, the algorithm must ensure the gradual transfer of control from the vehicle to the driver in the safest and most intuitive way for the driver in order to minimize the conflict between them.

The featured contributions of this paper can be summarized in the following aspects:

• A new cybernetic driver model that takes into account the visual and the neuromuscular aspects is proposed (Section 2).

The paper outline is as follows. The Driver-Road-Vehicle modeling is presented in Section 2. The online driver parameters identification approach is shown in Section 3 and the Driver-Automation shared control is designed in Section 4. Subsequently, in Section 5, the performance of the identification approach and the cooperative control scheme is validated in interactive simulation with the well-known SHERPA dynamic car simulator under real-world driving situations.

The vehicle lateral dynamics as result of interaction between tires and road surface can be represented by the bicycle model (Rajamani, 2011). The lateral tire friction forces for a vehicle with front steering can be expressed as F y f = 2 C f α f , F y r = 2 C r α r where the linear front and rear slip angles are given as follows: α f = δ − β − l f r v x , α r = l r r v x − β where δ is the wheel angle. The lateral side slip and yaw rate dynamics can be then given as (Sentouh et al., 2018), β ̇ = − 2 C f + C r m v x β + 2 C r l r − C f l f m v x 2 − 1 r + 2 C f m v x δ + 1 m v x f w r ̇ = 2 C r l r − C f l f I z β + − 2 C r l r 2 + C f l f 2 v x I z r + 2 C f l f I z δ + l w I z f w (1) where m is the mass of the vehicle, v _x is the longitudinal velocity, β is the drift angle at the center of gravity, I _z is the vertical moment of inertia of the vehicle, r is the yaw rate, l _f /l _r is the distance between the front/rear axle and the center of gravity of the vehicle, f _w is the lateral wind force having as its center of impact l _w away from the center of gravity. C _f and C _r are the lateral stiffness coefficients of the front and rear pneumatic respectively.

For lane tracking purpose, the vehicle position error y _L and the heading error ψ _L at a look-ahead distance l _p are taken into account in the control design. The dynamics of these variables are given as follows (Sentouh et al., 2018), ψ ̇ L = r − ρ c v x y ̇ L = β v x + l p r + ψ L v x (2) where ρ _c is the curvature of the road. To represent the haptic interaction between the human driver and the vehicle, the dynamics of the steering column is represented as follows, J s δ ̈ d = − B s δ ̇ d + T d + T a − T s (3) where δ _d is the steering wheel angle, J _s is the equivalent moment of inertia of the steering system, and B _s is its equivalent damping coefficient. T _d is the driver torque, and T _a is the assisting system torque. Where R _s is the reduction ratio between the steering wheel angle δ _d and the wheel angle δ. The self-aligning torque T _s is expressed as: T s = − 2 C f η t R s β + − 2 l f C f η t R s v x r + 2 C f η t R s 2 δ where η _t is the width of the tire contact.

It has been shown in the literature that the driver relies on two visual points to guide his vehicle on the road (McRuer et al., 1977; Donges, 1978), a far point allowing him to anticipate the evolution of the curvature and a near point which gives him a compensatory behaviour. θ near = y L l p + ψ L , θ far = D far r v x (4)

These points can be characterised by the two angles θ _near and θ _far as used in (Sentouh et al., 2009). The angle θ _far represents the angle between the vehicle heading and the tangent to the curve. As for θ _near , it represents the angle between the vehicle’s heading and the straight line connecting the vehicle’s centre of gravity and the point on the edge of the track, located at a distance of l _p as illustrated in Figure 2.In this paper, the driver model proposes to take into account the visual aspect by considering the anticipatory and compensatory behavior of the driver as used in (Sentouh et al., 2009) combined with the neuromuscular aspect as used in (Bi et al., 2015). The architecture of the proposed driver model is shown in Figure 3.

As used in (Sentouh et al., 2009), the visual model consists of three transfer blocks, G _a , G _c and G _d that represent respectively the anticipatory behaviour, the compensatory behaviour and the human processing delay time. In terms of anticipatory control, the driver predicts the future path of the vehicle and provides an anticipated steering input via the steering wheel before entering the curve. As for compensatory control, the driver adjusts the torque applied to the steering wheel by using visual information from the region near in front of the vehicle (the near visual angle θ _near ), in order to keep a safe lateral deviation from the centre of the lane. These transfers are modelled as (Sentouh et al., 2009): G a = k a , G c = k c T L s + 1 T I s + 1 , G d = e − τ d s (5) which k _a is the anticipation gain, T _L and T _I are the lead and delay time constants, respectively, and the gain k _c represents the proportional action of the driver with respect to the error of the near visual angle and τ _d is the delay time.

Based on the model used in the (Bi et al., 2015), the neuromuscular model consists of four transfer blocks, G _r , k _cs , G _rc and G _arm which represent respectively the reference model (feed-forward module), the stretch reflex controller, the co-contraction stiffness and the arm model. The reference model represents the angle-torque stiffness, which can provide a steering torque proportional to the desired angle. The co-contraction stiffness refers to the increase in the muscle’s intrinsic stiffness resulting from the activation of the muscle. The stretch reflex controller and the arm model represents the dynamics of muscle activation. These transfers are modeled as (Bi et al., 2015) and (Sentouh et al., 2009): G r = k r , k c s , G r c = ω c B r s + K s + ω c G arm = 1 T n s + 1 (6) which k _r is the feed-forward module, k _cs the co-contraction stiffness, K and B _r represent the stiffness and damping of the reflex, respectively, ω _c is the cut-off frequency of the first order filter and T _n is the time constant of the arm system.

In order to design a shared control, the following simplified architecture of the proposed driver model is used:

According to the simplified architecture shown in Figure 4, the following driving model is used: T d s = G arm s U d s (7) where K ₁(s), K ₂(s), K ₃(s) and U _d (s) are represented by the following transfer: K 1 s = G c G d G r + G r c + k c s K 2 s = G a G d G r + G r c + k c s K 3 s = − G r c + k c s U d s = K 1 s θ near s + K 2 s θ far s + K 3 s δ d s (8)

To resume, and from (7) the driver model can be expressed in the form of the following state representation T ̇ d t = a d T d t + b d u d t (9) where T _d (t) is the driver torque, a d = − 1 T n and b d = 1 T n are the matrices of the system, and u _d (t) is the driver system input u d t = k 1 t θ near t + k 2 t θ far t + k 3 t δ d t (10) where the new driver parameters used to design the shared controller are k ₁(t), k ₂(t) and k ₃(t) which represent the compensatory dynamics, the anticipatory dynamics, and the stiffness dynamics of the driver respectively. T _n is a constant, where its value is chosen as in (Sentouh et al., 2009).

In (8), K ₁(s), K ₂(s), and K ₃(s) are transfer functions (in the frequency domain), and in (10) are in the time domain where the latter are unknown (unknown temporal dynamics), so an online parameters identification using a Lyapunov (LP) approach is used to estimate them. This approach will be illustrated in the next section.

From the dynamics Eqs. 1–3 and Eq. 9 the driver-road-vehicle model can be formulated as follows: ∑ : x ̇ = A t x + B u + E d (11)

where x = β r ψ L y L δ d δ ̇ d T d T is the state vector, u = T _a is the control input, d = [f _w , ρ _c ] is the disturbance vector, and A t = a 11 a 12 0 0 a 15 0 0 a 21 a 22 0 0 a 25 0 0 0 1 0 0 0 0 0 v x l p v x 0 0 0 0 0 0 0 0 0 1 0 a 61 a 62 0 0 a 65 a 66 a 67 0 a 72 a 73 a 74 a 75 0 a 77 , B = 0 0 0 0 0 b 6 0 T , E = e 1 e 2 0 0 0 0 0 0 0 − v x 0 0 0 0 T . with: a 11 = − 2 C f + C r m v x , a 12 = 2 C r l r − C f l f m v x 2 − 1 , a 15 = 2 C f m v x R s , a 21 = 2 C r l r − C f l f I z , a 22 = − 2 C r l r 2 + C f l f 2 v x I z , a 25 = 2 C f l f I z R s , a 61 = 2 C f η t J s R s , a 62 = 2 C f l f η t v x J s R s , a 65 = − 2 C f η t J s R s 2 , a 66 = − B s J s , a 67 = 1 J s , a 72 = k 2 ′ t D far T n , k 2 ′ t = k 2 t v x , a 73 = k 1 t T n , a 74 = k 1 t T n l p , a 75 = k 3 t T n , a 77 = − 1 T n , b 6 = 1 J s , e 1 = 1 m v x , e 2 = l w I z

Note that A and E in Eq. 11 depend on the following measured, identified and bounded terms: v x , 1 v x , 1 v x 2 , k 1 , k 2 ′ , k 3 v min ≤ v x ≤ v max k 1 min ≤ k 1 ≤ k 1 max k 2 min ′ ≤ k 2 ′ ≤ k 2 max ′ k 3 min ≤ k 3 ≤ k 3 max With the approach of sector non-linearity (Wang and Tanaka, 2004), an exact T-S fuzzy model of Eq. 11 can be obtained with 2⁶ = 64 linear sub-systems. However, this vehicle T-S fuzzy model leads to not only expensive numerical burden for real-time implementation but also conservative control results. Here, to reduce significantly the complexity and the conservatism of the proposed control method, the following change of parameter is performed: 1 v x = 1 v 0 + 1 v 1 θ , θ min ≤ θ ≤ θ max (21) where: v 0 = 2 v min v max v min + v max , v 1 = 2 v min v max v min − v max . From (21) we obtain: v x = v 0 1 + v 0 v 1 θ − 1 , 1 v x 2 = 1 v 0 2 1 + v 0 v 1 θ 2

Note that the new parameter θ can be used to represent the variation of v _x between its lower and upper bounds, i.e., θ _min = −1, and θ _max = 1. Then, the following first-element Taylor’s series around zero is used to exploit the strong relationship between the speed-dependent terms: v x ≈ v 0 1 − v 0 v 1 θ , 1 v x 2 ≈ 1 v 0 2 1 + 2 v 0 v 1 θ (22)

Remark that v _x , 1 v x and 1 v x 2 in Eqs. 21,22 linearly depend on the new parameter θ. Substituting these expressions into Eq. 11, we obtain a vehicle model whose state-space matrices depend on θ , k 1 , k 2 ′ , k 3 . Using the sector non-linearity approach (Wang and Tanaka, 2004) leads to the following 16-rule T-S fuzzy representation of Eq. 11: ∑ : x ̇ = ∑ i = 1 2 ∑ j = 1 2 ∑ l = 1 2 ∑ m = 1 2 h i θ η j k 1 μ l k 2 ′ σ m k 3 A ijlm x + B u + E i w (23) where A _ijlm and E _i are calculated according to: i f i = 1 t h e n θ = θ min , i f i = 2 t h e n θ = θ max i f j = 1 t h e n k 1 = k 1 min , i f j = 2 t h e n k 1 = k 1 max i f l = 1 t h e n k 2 ′ = k 2 min ′ , i f l = 2 t h e n k 2 ′ = k 2 max ′ i f m = 1 t h e n k 3 = k 3 min , i f m = 2 t h e n k 3 = k 3 max

Moreover, the membeT-S fuzzy model are defined as follows: h 1 θ = θ max − θ θ max − θ min , η 1 k 1 = k 1 max − k 1 k 1 max − k 1 min , μ 1 k 2 ′ = k 2 max ′ − k 2 ′ k 2 max ′ − k 2 min ′ , σ 1 k 3 = k 3 max − k 3 k 3 max − k 3 min , h 2 θ = 1 − h 1 θ η 2 k 1 = 1 − η 1 k 1 μ 2 k 2 ′ = 1 − μ 1 k 2 ′ σ 2 k 3 = 1 − σ 1 k 3

To begin with, we define the performance output of Eq. 11 as z = C x = β r ψ L y L δ ̇ d T d T

Note that β, r, ψ _L and y _L represent lane tracking performance while δ ̇ d and T _d are included in z to represent driving comfort and driver effort attenuation respectively. To realize such a shared control scheme, we propose a control solution for the following design problem.

Problem 1. Consider the vehicle model system Eq. 23. Find a controller u such that:

• When w = 0, the closed-loop system is globally exponentially stable.

For shared control design, we consider the control law: u = ∑ i = 1 2 ∑ j = 1 2 ∑ l = 1 2 ∑ m = 1 2 h i θ η j k 1 μ l k 2 ′ σ m k 3 F ijlm x (25) where the control gains F _ijlm , i, j, l, m ∈ {1, 2}, are to be determined. The following result provides a control solution for Problem 1, where our analysis is conducted using the following Lyapunov function: V x = x T P x , P > 0 (26)

Theorem 2. Consider a T-S fuzzy system as described in Eq. 23. The time-varying controller Eq. 25 stabilizes system Eq. 23 while minimizing the performance index Eq. 24 if there exist positive definite matrix X, matrices M _ijlm , i, j, l, m ∈ {1, 2}, and positive scalars γ, ν, satisfying the following optimization:

Subject to ν x 0 T * X ≺ 0 (27) Γ ijlm ≺ 0 , i , j , l , m ∈ 1,2 (28) where Γ ijlm = H e A ijlm x + B M ijlm E i X C T M ijlm T * − γ I 0 0 * * − Q − 1 0 * * * − R − 1 (29)

Moreover, the feedback gains in Eq. 25 can be computed as follows: F ijlm = M ijlm X − 1 , i , j , l , m ∈ 1,2 (30)

In this subsection, we are going to presents the stability analysis is provided of the whole system.

Proof. Multiplying Eq. 28 with h _i (θ) ≥ 0, η _j (k ₁) ≥ 0, μ l ( k 2 ′ ) ≥ 0 and σ _m (k ₃) ≥ 0 and summing up for ∀ i , j , l , m ∈ 1,2 yields ∑ i = 1 2 ∑ j = 1 2 ∑ l = 1 2 ∑ m = 1 2 h i θ η j k 1 μ l k 2 ′ σ m k 3 Γ ijlm ≺ 0 (31)

For brevity, for any matrices of appropriate dimensions χ _i and Δ _ijlm , ∀i, j, l, m ∈ {1, 2} we denote χ θ = ∑ i = 1 2 h i θ χ i , Δ θ , k 1 , k 2 ′ , k 3 = ∑ i = 1 2 ∑ j = 1 2 ∑ l = 1 2 ∑ m = 1 2 h i θ η j k 1 μ l k 2 ′ σ m k 3 Δ ijlm

Using successively Schur complement lemma, it follows that inequality Eq. 31 is equivalent to ∑ ijlm + M θ , k 1 , k 2 ′ , k 3 T R M θ , k 1 , k 2 ′ , k 3 + ∇ E θ * − γ I ≺ 0 (32) where    ∑ ijlm = H e ( A ( θ , k 1 , k 2 ′ , k 3 ) X + B M ( θ , k 1 , k 2 ′ , k 3 ) ) , ∇ = X C T Q C X . Pre- and post-multiplying Eq. 32 with P = X ⁻¹, followed by the variable change M ( θ , k 1 , k 2 ′ , k 3 ) = F ( θ , k 1 , k 2 ′ , k 3 ) X , it follows that ∂ ijlm + F θ , k 1 , k 2 ′ , k 3 T R F θ , k 1 , k 2 ′ , k 3 + C T Q C P E θ * − γ I ≺ 0 (33) where ∂ ijlm = H e ( P ( A ( θ , k 1 , k 2 ′ , k 3 ) + B F ( θ , k 1 , k 2 ′ , k 3 ) ) ) . Pre- and post-multiplying Eq. 33 with x w T and its transpose, it follows that V ̇ x + z T Q z + u T R u < γ w T w (34) where V ̇ ( x ) is the time-derivative of the Lyapunov function candidate Eq. 26 along the trajectory of Eq. 23. From Eq. 34, two following cases are distinguished.

• For free-disturbance system (i.e., w = 0), V ̇ ( x ) < 0 , for ∀x ≠ 0, this means that system Eq. 23 with controller Eq. 25 is globally exponentially stable.

Since V(x) > 0, it follows from (35) that I < x 0 P x 0 + γ w 2 2 (36) where w 2 2 denotes the L2-norm of w. By Schur complement lemma, it follows from Eq. 27 that x ₀ Px ₀ < ν Hence, note from Eq. 36 that Eq. 24 can be minimized by minimizing γ and ν. This concludes the proof.

This subsection is composed of two parts, the first one consists in the comparison between the proposed identification approach based on Lyapunov (LP) and the RLS approach, and the second part consists in the validation and analysis of the driver behaviour model in different tracks.

In this subsection, we are going to made a comparison between our shared controller and the shared controller proposed in (Nguyen et al., 2017), the latter uses a shared control based on a driver model with constant parameters for a lane-keeping scenario (these parameters are obtained offline). This comparison is made in the case of an obstacle avoidance test in highway track, we placed three obstacles and asked three different drivers to avoid them by changing lanes in shared control mode. The test results are shown in Figure 12. It can be seen that the three drivers found it difficult to overtake the obstacle using the controller with constant driver parameters, where the driver and controller torques are high, compared to the proposed shared controller that uses the driver model with estimated parameters online, the three drivers can overtake the obstacle easily without applying high torque. This can be explained by the fact that if we use the same model with constant parameters for obstacle avoidance, the model may be invalid. This result confirms the results obtained in the identification phase shown in Figure 10, Figure 11, where it shows that the driver’s behavior changes depending on the situation encountered. All these results show the effectiveness of the proposed method in adapting the shared control to different scenarios and different drivers.

In order to reinforce the results already obtained, an objective evaluation based on metrics to analyse the interaction of the driver with the system and the quality of control was done for the first driver. The interaction is evaluated through the torque exchanged and the angular velocity of the steering wheel during the test phase using the tow controllers, the controller based on constant driver parameters (Shared_CDP) and the proposed controller based on online driver parameters (Shared_ODP). To this end, four indicators were analyzed:

• The total steering effort provided by the driver StED (N.m2 for Steering Effort) during the period of experimentation (T _Ex ):

The results of the objective analysis are presented in Figure 13. As we can see, the indicator of the effort provided by the driver and the system using a controller that does not take into account the driver’s behavior (Shared_CDP) is high compared to that of (Shared_ODP), this result shows that the proposed controller minimizes the effort applied by the driver in shared control mode and minimize the system effort in order to avoid saturation and overheating of the steering system motor. In addition, even if we compare the conflict indicator, we can clearly see that the (Shared_ODP) minimizes the conflict between the system and the driver in a dynamic way compared to the one that does not take the driver’s behavior in consideration (Shared_CDP). Comparing the last indicator, which represents the driving workload, it can be seen that the proposed controller minimizes the workload compared to the (Shared_CDP). So the proposed controller will allow us to minimize the driver effort, the system effort, the conflict between the driver and the system, and the workload on the steering wheel in shared control mode. The obtained results have strongly demonstrated the effectiveness of the proposed shared control method in the attenuation of driver effort and conflict between the lane keeping system and the driver.