Robust Nonlinear Predictive Control for Semiautonomous Ground Vehicles

Transcription

1 14 American Control Conference (ACC) June 4-6, 14. Portland, Oregon, USA Robust Nonlinear Predictive Control for Semiautonomous Ground Vehicles Yiqi Gao, Andrew Gray, Ashwin Carvalho, H. Eric Tseng, Francesco Borrelli Department of Mechanical Engineering, UC Berkeley, USA; Ford Motor Company, USA; Abstract This paper presents a robust control framework for lane-keeping and obstacle avoidance of semiautonomous ground vehicles. It presents a systematic way of enforcing robustness during the MPC design stage. A robust nonlinear Model Predictive Controller (RNMPC) is used to help the driver avoid obstacles and track the road center line. A forceinput nonlinear bicycle vehicle model is developed for the RNMPC control design. A robust invariant set is used in the RNMPC design to ensure robust satisfaction of state and input constraints in the presence of disturbances and model errors. Simulations and experiments on testing vehicles show the effectiveness of the proposed framework. Index Terms vehicle safety; uncertain dynamics; robust control; active safety; autonomous vehicles; robust nonlinear MPC I. INTRODUCTION Autonomous cars and smart active safety systems on cars have been drawing increasing attention recently [1], [], [3]. Such systems usually involve real-time generation and tracking of feasible trajectories for the vehicle. This is challenging when the vehicle travels at the limit of its handling capability. There are two main difficulties. First, trajectories generated by using oversimplified models violate system constraints, while computing trajectories using highfidelity vehicle models is computationally demanding. Second, uncertainties from a variety of sources might prevent the vehicle from following the desired paths and satisfying the safety constraints. Such uncertainties include measurement errors, friction coefficient estimation, driver behaviour and model mismatch. Because of its capability to systematically handle system nonlinearities and constraints, work in a wide operating region and close to the set boundary of admissible states and inputs, Model Predictive Control (MPC) is an attractive method to generate feasible trajectories and track them [4]. Parallel advances in theory and computing systems have enlarged the range of applications where real-time MPC can be applied [5], [6], [7], [8]. Previous work by the authors [9], [1], [11] have accounted the avoidance of pop-up obstacles through a hier- {yiqigao,ajgray,ashwinc,fborrelli}@berkeley.edu. htseng@ford.com. This material is based upon work supported by the National Science Foundation under Grant No. [13933]. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. archical framework. The present work builds on the past success [4] and presents a systematic way of enforcing robustness during the MPC design stage. A robust nonlinear MPC framework [1], [13], [14], [15], [16], [17] is used in this paper. In particular we follow the Tube approach presented in [17] for nonlinear system where the inputs enter linearly in the state update equations. The basic idea is to use a control law of the form u = ū+k(ξ ξ) where ū and ξ are the nominal control input and system state trajectories. For a given linear controller K, a robust invariant set is computed for the error system. The invariant set is used to bound the maximum deviation of the actual states from the nominal states under the linear control action K. A nominal MPC optimizes ū and ξ with tightened state and input constraints. The tightening is computed as a function of the bounds derived from the robust invariant set. In the first part of the paper we propose a force-input nonlinear bicycle vehicle model. In the second part we show how to compute the robust invariant set utilizing the lipschitz constant of the nonlinear system. In the third part we use the robust invariant set computed off-line to tighten the state and input constraints for the nominal nonlinear MPC problem which computes obstacle-free trajectories and corresponding input sequences. Experiments in different scenarios show the effectiveness of the proposed approach. The paper is structured as follows. In section II the force-input nonlinear vehicle model is developed. Section III presents the Robust MPC framework and the computation of the robust invariant set. Section IV presents the safety constraints and section V formulates the Robust MPC problem. Finally, in section VI we present the experimental results showing the performance of the proposed controller. More details on MPC implementation and more test results can be found in [18]. II. MODELING In this section we present the mathematical models used for control design. A. Bicycle vehicle model Consider the vehicle sketch in Figure 1. We use equations (1) to describe the vehicle motion within the lane. In (1), m and denote the vehicle mass and yaw inertia, respectively, a and b denote the distances from the vehicle center of gravity to the front and rear axles, respectively. ẋ and ẏ denote the /$ AACC 4913

2 Fig. 1. Modeling notation depicting the forces in the vehicle body-fixed frame (F x and F y ), the forces in the tire-fixed frame (F l and F c ), and the rotational and translational velocities. The relative coordinates e y and e ψ are illustrated on the sketch of the road as well as the road tangent ψ d. Point P is the center of percussion as will be discussed in section II-B. vehicle longitudinal and lateral velocities, respectively, and ψ is the turning rate around a vertical axis at the vehicle s center of gravity. e ψ and e y in Figure 1 denote the vehicle orientation and lateral position, respectively, in a road aligned coordinate frame and ψ d is the angle of the tangent to the road centerline in a fixed coordinate frame. s is the vehicle longitudinal position along the desired path. mẍ = mẏ ψ + F xf + F xr, mÿ = mẋ ψ + F yf + F yr, ψ = afyf bf yr, ė ψ = ψ ψ d, ė y = ẏ cos(e ψ ) + ẋ sin(e ψ ), ṡ = ẋcos(e ψ ) ẏsin(e ψ ), (1a) (1b) (1c) (1d) (1e) (1f) Assumption 1: Only the steering angle at the front wheels can be controlled. i.e., δ f = δ and δ r =. The longitudinal and lateral tire forces F l and F c are given by Pacejka s model [19]. The nonlinear tire force model increases the computational burden of real-time model-based control techniques. We will approach this issue as follows. For the front wheels, since both slip angle and slip ratio can be controlled, it is valid to assume that any force within the friction circle can be achieved. Thus we will make the front tire forces F xf and F yf as the inputs to the model. For the rear wheels, however, the slip angles are not directly controllable due to Assumption 1 and are functions of the vehicle states. Thus the rear tire forces cannot be treated in the same way as the front tire forces. In the following subsections we will discuss how to simplify the rear tire force model. B. Center of percussion The center of percussion (CoP) is a point along the vehicle longitudinal axis where the rear tire lateral forces do not affect its dynamic evolution ([], [1]). As discussed in [1], the rear tire lateral forces have two effects on dynamics, a constant lateral acceleration on the vehicle body and an angular acceleration around the CoG. At the CoP these two effects cancel each other and thus ẏ p is not influenced by the rear tire forces. Figure 1 illustrates the concept of CoP. Note that ẋ p = ẋ. To find the position of the CoP, we refer to the following dynamic equations ẏ p = ẏ + p ψ ÿ p = ÿ + p ψ By substituting (1b) and (1c) into (b) we obtain (a) (b) ÿ p = ẋ ψ + ( m + ap )F yf + ( m bp )F yr. (3) When p = mb, the F yr term in (3) vanishes. C. Simplifying rear tire forces The use of the CoP eliminates the rear tire force in ÿ p. However, the rear tire forces are still present in the longitudinal and yaw dynamics ẍ p and ψ. We will use a linearized tire model for these terms. Let β r denote the braking/throttle ratio of rear tire (F lr = β r µf zr ), with β r =1 corresponding to max throttle and -1 max braking. In moderate braking, the resultant lateral tire force F c as a function of slip angle α has small sensitivity with respect to β r. Thus a constant linear gain can be used over β [.5 ] and over β [.5]. The remaining rear tire forces in (1a) and (1c) are modeled as: F lr = β r µf zr F cr = C r α r The slip angles α are approximated as follows: α f = ẏ + a ψ ẋ δ, (4a) (4b) α r = ẏ b ψ. (5) ẋ Remark 1: The same linearization could be applied to the F yr term in (1b). We decided to use CoP to eliminate it (Section II-B) to reduce the error introduced by the linearization. D. Force-input nonlinear bicycle model In summary, the nominal vehicle model (1) used for control design is rewritten as in (6) ÿ p = (a + b)µf zf β xf mb ẍ p = µf zf β xf m ψ = aµf zf β yf ė ψ = ψ ψ d, ė y = ẏ p + ẋ p e ψ, ṡ = ẋ p ψẋ p, (6a) + µf zrβ r m + ψẏ p ψ p, (6b) bc rẏ p ẋ p + bc r(b + p) ψ ẋ p, (6c) (6d) (6e) (6f) In (6) the states are ξ = [ẏ p, ẋ p, ψ, e ψ, e y, s] T R 6 1, and the inputs are u = [β xf, β yf, β r ] T R 3. β xf and β yf are the normalized longitudinal and lateral tire forces on the front wheels. 4914

3 III. ROBUST MPC AND INVARIANT SET A. Tube-based robust MPC for nonlinear systems In this section we outline the framework used to develop the robust model predictive controller in section V. We follow a notation similar to [13]. Consider the system: ξ k+1 = Aξ k + g(ξ k ) + Bu k + w k. s.t. ξ k Ξ u k U (7a) (7b) (7c) Where g( ) : R n R n is a nonlinear Lipschitz function and w k W R n is an additive disturbance. We assume that the matrix pair (A, B) is controllable. The control problem is divided into two parts. First a feedforward control input is computed for the system (7). System (7) with the computed feedforward input and a zero disturbance sequence will be called the nominal system. Then, we design a state feedback controller which acts on the error between the actual state of system (7) and the predicted state of the nominal system. Next we formalize the aforementioned ideas. We denote the N-step control sequence and the N- step disturbance sequence for system (7) as u = {u, u 1,..., u N 1 } and w = {w, w 1,..., w N 1 }, respectively. Let Φ(k, x, u, w) denote the solution of (7) at time k controlled by u when x = x. Furthermore, let Φ(k, x, ū) denote the solution of the nominal system: ξ k+i+1 = A ξ k+i + g( ξ k+i ) + Bū k+i, i =, 1,..., N 1. (8) Let e k = ξ k ξ k be the error between the states of system (7) and the nominal system. Let the input u k to system (7) be: u k = ū k + û(e k ), (9) where û(e k ) : R n R m. We define the error dynamics as: e k+1 = Ae k + Bû(e k ) + (g(ξ k ) g( ξ k )) + w k. (1) The authors of [13] proved the following result. Proposition 1: Suppose that Z is a Robust Positively Invariant Set[13] of the error system (1) with control law û( ). If ξ k { ξ k } Z, then ξ k+i { ξ k+i } Z for all i > and all admissible disturbance sequences w k+i W. Proposition 1 states that if the state ξ of system (7) starts close to the nominal state ξ, the control law (9) will keep the state trajectory Φ(k, x, u, w) within the robust positively invariant set Z centered at the predicted nominal states Φ(k, x, ū) for all admissible disturbance sequence w: ξ { ξ } Z ξ k { ξ k } Z w k W, k. (11) Proposition 1 also suggests that if a feasible solution can always be found for the nominal system (8) subject to the tightened constraints Ξ = Ξ Z, Ū = U û(z), (1) then the control law (9) will ensure constraint satisfaction for the controlled uncertain system (7) [15]. Remark : For general nonlinear systems, the controller and invariant set pair (û( ), Z) is usually hard to find. In the following section we will propose an algorithm for computing (û( ), Z) for a system in the form of (7), whose dynamics consist of a linear term Aξ + Bu and a small nonlinear term g(ξ). B. Computation of the robust invariant set The main idea is to bound the nonlinear term (g(ξ) g( ξ)) in (1) by using the Lipschitz constant of g( ), and treat it as part of the disturbance. We use the -norm over R n. The nonlinear function g(ξ) is Lipschitz in the set Ξ with respect to ξ if L > such that g(ξ 1 ) g(ξ ) L ξ 1 ξ, ξ 1, ξ Ξ. (13) The smallest constant L satisfying (13) is called the Lipschitz constant. Let L(Ξ) be the Lipschitz constant of g( ) over Ξ. From (13) we know ξ, ξ Ξ and e = (ξ ξ) E g(ξ) g( ξ) L(Ξ) max e E e, (14) where E is a subset of Ξ containing {}. The inequality in (14) defines a box B(E) = {x R n x L(Ξ) max e } and is used to bound the term (g(ξ k ) e E g( ξ k )) in (1). Equation (1) is then written as: e k+1 = Ae k + Bû(e k ) + w k, (15) where w k W = W B(E). Proposition : For ξ, ξ Ξ, If Z is a robust positively invariant set for system (15) with a control law û( ), then Z is a robust positively invariant set for system (1) with the same control law û( ). Proposition is true because all possible values of (g(ξ) g( ξ)) in (1) lies inside B(E). Since we assumed the matrix pair (A, B) to be controllable, there exists a stabilizing linear feedback gain K such that (A + BK) is Hurwitz. Algorithm 1 is used to compute the robust positively invariant set Z associated with the gain K. If Algorithm 1 terminates in finitely many iterations i t, Algorithm 1 Computation of Z 1: Ω {} : W W 3: i ; 4: repeat 5: i = i+1 6: W = W B(Ωi ); 7: Ω i = Reach fa (Ω i 1, W) Ω i 1. 8: until Ω i == Ω i 1 9: Z Ω i then Z = Ω it is the minimal positively invariant set for the uncertain system (15) []. We choose the stabilizing state feedback gain K to be the infinite horizon LQR solution KLQR for system (A, B). Therefore the controller (9) becomes: u k = ū k + K LQR(ξ k ξ k ). (16) 4915

4 C. Model reformulation The vehicle model (6) can be written in a similar form of (7): ξ = Aξ + g(ξ) + Bu + G + w, (17) The constraints (19)-(1) are compactly written as, h(ξ, u, w), () where is the zero vector with appropriate dimension. In (17), all nonlinear terms are collected in g( ) : R n R n. V. ROBUST PREDICTIVE CONTROL DESIGN The matrices in (17) are given in (18). In this section we present the MPC formulate for the lane keeping and obstacle avoidance problem. At each sampling ẋ p time instant an optimal input sequence is computed by A = bc r bc I ẋ p r (p+b) solving a constrained finite time optimal control problem. I ẋ p 1, The computed optimal control input sequence as well as the corresponding predicted vehicle state trajectory are stored 1 ẋ p as the nominal input and state trajectories, ξ and ū. At the 1 next time step the optimal control problem is solved staring ψ from new state measurements. An algorithm computes the ψẏ p ψ p augmented input u = ū + Ke at each sampling time. The bc r (ẏ p ẋ p (p+b) ψ ẋ p ) MPC and the computation of u can be executed at different g(ξ) = ẋ p( ẋ p+ ẋ p), (18a) sampling times. We discretize the system (17) with a fixed sampling time ẋ p e ψ T s to obtain, µf zf m + x aµf k+1 = f n ( x k, ū k ), (3) zf mb µf zf µf m zr m aµf B = zf and the nominal optimization problem with tightened con-, G = straints is formulated as in (4). In (4), t denotes the ψ d. current time instant and ξ t+k,t denotes the predicted state at time t + k obtained by applying the control sequence ū = {ū t,t,..., ū t+k,t } to (3) with ξ t,t = ξ(t). H p and H c denote the prediction horizon and control horizon, respectively. We (18b) denote by H i the input blocking factor to hold the ū k constant for a length of Hi. This reduces the number of Note ẋ p is rewritten as ẋ p = ẋ p + ẋ p. w represents an optimization variable which is useful in real-time implementation. The safety constraints () have been imposed as soft additive disturbance. We follow the approach described in section III-B to compute the positively invariant set Z. constraints, by introducing the slack variable ε in (4b) and IV. SAFETY CONSTRAINTS (4c). Q, R, S and λ are weights of appropriate dimension penalizing state tracking error, control action, change rate of control, and violation of the soft constraints, respectively. The main objective of the safety system proposed in this paper is to keep the vehicle on the road while avoiding obstacles. In this section, we detail the constraints imposed on the vehicle states and inputs to guarantees safe maneuvers. Road boundary constraint: we constrain the CoG of the vehicle and shrink the road bounds to take the vehicle width into account. The road boundary constraints are written as ē ymin e y ē ymax, (19) where ē ymin and ē ymax are obtained from the tightened state constraint Ξ. Obstacle avoiding constraint: consider ellipsoidal obstacles in the form ( s s obs a obs ) + ( ey y obs b obs ) 1. The obstacle avoiding constraints are written as ( s s obs s Z ) + ( e y y obs y Z ) 1, () a obs b obs where s Z and y Z are the projection of the invariant set onto s and e y axes. Slip angle constraint: we constrain the tire slip angles to belong to the linear region of the tire forces. α min α α max. (1) min ū,ε H p 1 k= ξ t+k,t ξ ref Q + ū t+k,t R ū t+k,t S + λε s.t. ξ t+k+1,t = f n ( ξ t+k,t, ū t+k,t ), k =, 1,..., H p 1 (4a) (4b) h t ( ξ t+k,t, ū t+k,t ) 1ε, k = 1,..., H p (4c) ε, ū t+k,t = ū t+k,t + ū t+k 1,t, (4d) (4e) ū t+k,t Ū, k =, 1,..., H c 1 (4f) ū t+k,t Ū, k =, 1,..., H c 1 (4g) ū t+k,t =, k = H c,..., H p (4h) ū t 1,t = ū(t 1), ξ t,t = ξ(t), (4i) (4j) Remark 3: Because of the system nonlinearities and time varying constraints, it is difficult to guarantee the feasibility of the nominal nonlinear MPC under all possible scenarios. 4916

5 However, if the nominal MPC is feasible, the vehicle is guaranteed to keep the actual trajectory within the robust bounds under any admissible disturbance. VI. EXPERIMENTAL RESULTS The experiments have been performed on two test vehicles with different surface conditions. We first tested the controller on Test Vehicle A at a test center equipped with icy and snowy handling tracks. Test Vehicle A has a mass of 5 Kg and a yaw inertia of 3344 Kg/m. It was equipped with an Active Front Steering (AFS) and Differential Braking system which utilizes an electric drive motor to change the relation between the hand steering wheel and road wheel angles. We used an Oxford Technical Solution R (OTS) RT3 sensing system to measure the vehicle position and orientation in the inertial frame and the vehicle velocities in the vehicle body frame. The OTS RT3 is housed in a small package that contains a differential GPS receiver, Inertial Measurement Unit (IMU) and a DSP. The IMU includes three accelerometers and three angular rate sensors. The DSP receives both the measurements from the IMU and the GPS, utilizes a Kalman filter for sensor fusion, and calculates the position, orientation and other states of the vehicle. The controller was run in a dspace R Autobox R system, equipped with a DS15 processor board and a DS1 I/O board. The sensor, the dspace R Autobox R, and the actuators communicate through a CAN bus. Test Vehicle B was used to test the controller on asphalt tracks. It has a mass of 1544 Kg and a yaw inertia of 3477 Kg/m. Test Vehicle B has similar instrumentation as Test Vehicle A with two main differences. First, it uses an Electric Power Steering system. Second, the longitudinal dynamics are controlled by a low level controller which achieves required longitudinal accelerations. The parameters used in the experiments are summarized in Table I. TABLE I REAL-TIME DESIGN PARAMETERS Param. Value Units Param. Value Units u max [,.5, ] [ ] H p 1 - u min [.5,.5, ] [ ] H c 8 - u max [,, ] [ ]/sec H i - u min [,, ] [ ]/sec T s 1 ms Q (1,, 5, 1) - R (1, 1, 1) - S (1, 1, 1) - Y [m] Actual path Nominal path Tube bound Center line X [m] Fig.. Experimental result: Test Vehicle A enters the maneuver at 8Kph on a snow track (µ.3). The dashed black line and blue line are the nominal and actual vehicle trajectories respectively. The green dot-dashed lines indicate the robust bounds around the nominal trajectory. The actual vehicle path is seen to be very close to the nominal one and within the robust bounds. δf [deg] Tbfl [Nm] Tbrl [Nm] Time [s] Fig. 3. Experimental result: The steering and braking input during the experiment of Fig. B. Robust performance against friction coefficient In this section we test the Test Vehicle A on an ice track with a tire-road friction coefficient around.1. The controller is set up for a nominal µ of.3, as it is on a snow track. The vehicle avoids the obstacle and tracks the center line at a speed of 35 kph. Figure 4 shows the trajectory of the vehicle during the maneuver and Figure 5 shows the inputs. C. Obstacle avoidance on high µ surface This section shows the test results with Test Vehicle B on an asphalt track. The vehicle travels on a two lane road. It avoids an obstacle by partially moving into the other lane. It then turns back and follows the original lane center. The behavior of the controller is consistent with different test vehicles on different surfaces. D. Computational time of the controller One advantage of the proposed controller is its low computational burden compared to other robust approaches we have A. Obstacle avoidance at high speed on low µ surface In this section we show the controller s ability to control the Test Vehicle A to avoid an obstacle on the snow track at high speeds. Figure show the trajectory of the vehicle, where the entering speed is 8kph. The dot-dashed lines denotes the bounds of the invariant set. The vehicle successfully avoids the obstacle and returns back to the lane center. Figure 3 shows the input from the controller in the same trial. We observe a brief braking at the beginning of the maneuver similar as we see in the simulation. Y [m] Actual path Nominal path Tube bound Center line X [m] Fig. 4. Experimental result: Trajectory of Test Vehicle A on an ice track. The vehicle enters the maneuver at 35Kph. The actual µ on the track is.1, while the controller is set up for µ =.3 on snow track. 4917

6 δf [deg] Tbfl [Nm] Tbrl [Nm] Time [s] δf [deg] Tbfl [Nm] Tbrl [Nm] Time [s] Fig. 5. Experimental result: The steering and braking input during the experiment of Fig 4. Fig. 7. Experimental result: The steering and braking input during the experiment of Fig 6. Y [m] 6 4 Actual path Tube bound Center line Lane marker X [m] Fig. 6. Experimental result: Test Vehicle B enters the maneuver at 55Kph on a two lane road. It avoids the obstacle by moving into the adjacent lane and gets back to the original one afterwards. tried in our lab. Since the invariant set was computed offline, the on-line computational burden of the robust controller is almost the same as the nominal MPC. For the four tests reported previously, the average processing time was 59.6 ms and the maximum was 71.6 ms, both below the sampling time of 1ms. VII. CONCLUSIONS This paper presents a robust control framework for lane keeping and obstacle avoidance. 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