Integrated Vehicle Control via Coordinated Steering and Wheel Torque Inputs

Transcription

1 Proceedings of the Aerican Control Conference Arlington, VA June 25-27, 21 Integrated Vehicle Control via Coordinated Steering and Wheel Torque Inputs Sean Brennan Andrew Alleyne ~ Departent of Mechanical & Industrial Engineering University of Illinois, Urbana-Chapaign Urbana, IL 6181 Abstract A controller was developed to govern the lateral position of a highway-speed vehicle using frequencyweighted coordination of front steering and torque inputs. The MISO design proble was recast as a SISO approach by using a cascaded design technique: the first step deterined the relative contribution of each control input as a function of frequency; secondary design steps utilied classical SISO approaches. For the vehicle control proble, the torque steering inputs were designed to act only as high-frequency inputs, while standard front steering was weighted for DC and low-frequency inputs. This controller was then tested on an experiental vehicle syste. 1. Introduction The independent use of either front steering, rear steering, and differential torque steering to control the yaw rate of a vehicle is a well-studied area of vehicle control. However, two issues are driving the vehicle research toward coordinated use of steering inputs: safety considerations have becoe a priary selling point with new vehicles; and newer vehicle designs such as electric vehicles are offering increasing opportunities for coordinated steering control. Historically, the coordinated use of steering inputs is not a new concept. Several authors, notably [1], have established that independent control of several steering inputs can grant the vehicle syste certain desirable properties such as coplete dynaic separation between side-slip and yaw rate states. Other studies, such as [2], have investigated the wheel usage during coordinated aneuvers using the concept that axiu tire-force usage is a true easure of the syste stability argin. Unfortunately, coordinated-steering investigations in general rely on the assuption that the syste states such as side-slip angle and wheel forces are readily available, while in practice they are usually only estiated with a large argin of error. The scope of the vehicle control literature is quite extensive, and the reader is referred to review articles [3] and [4] for appropriate suaries of the field. The intent of this article is to present a ethodology for dealing with a steering coordination task, and the control ethodology has been intentionally liited to a relatively siple linear odel. Other authors have appropriately dealt with the syste nonlinearities, ost notably in [5], [6], [7], and [8]. Both wheel steering and wheel torque ethods of driving are known to exhibit strong non-linearities as the steering inputs saturate, and any authors have dealt with these nonlinearities in the structure of the control algorith. Rather than deal with the nonlinearities either through robust analysis or adaptation, a better integration of existing control strategies was sought. The controller ethodologies considered are the linear PQ design ethod developed in Schroeck and Messner [9]. The approach in [9] is considered priarily because it takes advantage of key structural properties of the vehicle dynaics and has deonstrated perforance in Dual-Input Single-Output (DISO) systes such as hard-disk head controllers. This controller is exained in this study both in siulation and on an experiental scale vehicle given center-of-gravity lateral position as the sole feedback state. The reainder of the article is organied as follows: First, an overview is presented of the vehicle dynaics under study as a Multi-Input Single-Output (MISO) controller design proble. The next section introduces the MISO design technique referred to as the PQ controller. The third section exaines the specific ipleentation of the MISO controller on the vehicle syste, and the fourth section presents ipleentation results on an experiental vehicle. A conclusion then suaries the ain points of the paper. 2. Vehicle Dynaics Modeling of the vehicle dynaics is accoplished by fixing a coordinate syste to the center of gravity (CG) of the vehicle and applying Newton's equations. Roll, pitch, bounce, and deceleration dynaics are neglected to siplify the vehicle dynaics to two degrees of freedo: the lateral position states and yaw angle states. The odel is further siplified by assuing that each axle shares the sae steering angles, and that consequently each wheel produces the sae wheel angle steering forces. The resulting linear dynaic odel, known as the bicycle odel, conceptually atches a bicycle constrained to inplane otion. The use of this odel is explained in detail by Dugoff, Fancher, and Segel [ 1]. Although the bicycle l Corresponding author: alleyne@uiuc.edu /1/$1. 21 AACC 7

2 _ at" odel is relatively siple, it has been verified as a good approxiation for full-sie vehicle dynaics when accelerations are liited to.3 g's [ 11 ]. This study restricts the analysis to consider only the linear vehicle odel. This study does not investigate tire saturation behavior; in such cases a odel-switching ethod could be used when tire forces saturate as easured by the ABS or siilar wheel slip sensor. Additional assuptions ust be ade to extend the traditional linear vehicle odel to the case where a differential torque coand is used to steer the vehicle. The drive torque can be separated into two coponents: the steady-state driving torque and the transient steering torque. The steady-state driving torque is assued to not produce a steering oent and not to affect the transient vehicle dynaics. The steering torque inputs are assued to be "odd" about the longitudinal axis and are transitted via the front axle only. That is, if the right-hand side of the vehicle is sent a positive steering torque coand, the lefthand side of the vehicle is sent a steering torque of equal and opposite sign. With the above assuptions, a state-space odel can be obtained using the following ethodology: First, the vehicle dynaics are written in state-space for with the tire forces acting as inputs to the syste. We then solve for the tire forces as functions of the vehicle's lateral velocity and yaw rate, and control inputs fro the fro front steering and wheel torque inputs, thus copleting the statespace representation. As a sign convention, the Society of Autootive Engineers standard coordinate syste convention is used with the -axis pointing into the road surface. The wheel torque that tends to spin the vehicle in the positive yaw direction, shown in Fig. 3, will be considered positive. Z L_~~ direction of positive... yaw rate Figure 1" Vehicle notation. The state equations for the controller design are fored in ters of error states as the vehicle is forced to follow a trajectory: Error Ref. ~States[ H ~. Measured State,"1 Controller Vehicle State Trajectory _ Trajectory Figure 2: Error states. The error state equations are as follows: 2=A.x+B.u A= B= 1 Caf + C C af ~. U _ a.caf -b.cat. I.U o o at" a.caf Ca~'a -C.b d. ~. I I.r o +c ar (1) o a " Caf - b " Ca, r.u 1 -b.c a 2.C +b 2 ~r of.c I I.U With states and control inputs: y = lateral position error [] g/= yaw angle of the vehicle w.r.t, ground [rad] AT, ~ = steering torque input and front steering angle. and plant paraeters: = vehicle ass [kg] (6.2) /~ = orn. of inertia, -axis [kg- 2] (.15) U = constant longitudinal velocity [/s] (4.) CaT, Car = f. and r. wheel cornering stiffness, (39,6) a, b- length of f, r axle fro C.G. [], (.137,.22) r- wheel radius [], (.2995) d = dist. fro centerline to wheel [] (.115) The values in parenthesis represent the nuerical values easured for the scale experiental vehicle. If the torque input is ignored, the resulting linear state-space odel agrees with published dynaics fro [12], aong others within a non-diensional fraework [13]. The transfer functions Gi(s) fro front, rear, and differential torque steering to lateral position are given fro the transforation Gi(s ) = C(sI-A)-lBi. The characteristic equation is given as den, defined as (% *%! a 1 = U + %cl 2 a = I U 2 The transfer functions are: den = s 2 + a 1 s + a o (2) (% a2 + Ca, rb2 / I IU

3 5r( ) CafCbL Caf. s 2 n t- "S-t- or C af C or L y (s / U I I _ Z Z den _ C or " S 2 Jr- CafC UI al C CL or af or Z a. -b.c.s+~. +C y (s) r UI or ri _ Z Z AT(s) den The lateral position error given all three separate inputs is given by: "S-- y(s) :- Gl(S).bll(S)-t-G2(s).u2(s)-t-G3(s).u3(s) (4) A clear goal of the MISO vehicle controller proble is to coordinate the inputs to prevent tire force saturation. In particular, coordination between the wheel steering and torque steering control inputs is especially critical; if the front steering inputs are opposing the torque steering inputs, then the tire forces are unnecessarily high. For this reason, a control design ethodology was sought that specifically accounted for coordination between input channels. 3. MISO Controller Design Methods To accoodate a frequency weighting constraint between the steering and torque inputs, a MISO controller design ethodology was chosen that was originally presented in Schroeck and Messner [9]. To explain this ethod, a siple DISO plant is first considered with three controllers as shown in Figure (3): Controller I Figure 3: DISO plant. Z Plant To design the controller, first the ratio of the transfer functions G1 and G 2 are taken as a new plant P as shown in Figure 4. A controller Q is then designed to shape P such that a desired frequency roll-off is achieved. High values of the loop gain PQ(je) indicate that input 2 doinates at this frequency while low values indicate that input 1 doinates. Thus, if Gl is a fine, high-frequency actuator and G 2 is a coarse, low-frequency actuator, a large PQ loop gain is desired at low frequencies and low loop gain is!1 (3) desired at high frequencies. Thus the frequency weighting proble is siply re-cast as a controller loop-shaping proble. Selection of Q decides the relative frequency separation of the parallel subsystes fored by C1G 1 and C262. Q p Figure 4: DISO plant. After Q is designed, this controller transfer function is then broken up between C2 and C1 such that each controller is realiable. If Q is realiable, then a natural choice is C~ = 1, C2=Q. With C1 and C fixed, the controller design proble becoes a SISO design proble in ters of selecting Co to stabilie Gsiso as shown in Figure 5. Again, any design technique can be used to design Co, but loop-shaping procedures are quite easy to ipleent. I '... i,... Gs]so Figure 5: asiso plant and Co controller. The extension of the PQ design technique is easily extended to higher order odels, however for the vehicle steering coordination task, a two-input coordination task for wheel steering and torque steering for a front-wheel drive vehicle serves as a good exaple for this technique. 4. Vehicle Controller Design To deonstrate the design of a coordinated steering PQ controller, a vehicle plant is considered where a frontwheel-drive vehicle's lateral position on the road is controlled. The feedback is easured lateral position error as easured fro a sensor ounted at the C.G. of the vehicle. Both wheel steering and torque steering are available as steering inputs, but both ust act through the sae tires, thus coordination is necessary for best perforance. The vehicle odel is chosen as the easured vehicle paraeters substituted into the state space equations of equation (1). The Bode plots of the syste response is shown in Figure 6.

4 ' ~ ~ " ~ / ~ Front Torque ~ ' ~ ~ Figure 6" Bode plots fro front and torque steering inputs to lateral position f Figure 8" Bode plot of P(s)Q(s). To coordinate the two inputs, the front steering input is chosen as the course input (G2), while the torque steering input is chosen as the fine input (G1). Therefore, for low frequency inputs the front wheel steering will be priarily be active, while for high-speed inputs the torque steering inputs will becoe active. The Bode plot of P(s)= C2(s)/Cl(s) is given in Figure 7. 4O 2 For this exaple, Q(s) was chosen as: Q(s)= S 2 n t s + 1 Figure 8 shows the Bode plot cascaded syste P(s)Q(s) revealing the desired weighting. Since Q(s) is realiable, the controller Q was separated as Cl(S) = 1 and C2(s) = Q(s). Using these values for Cl(s) and C2(s), Gsiso(S) is fored and the Bode plot of the uncopensated syste is shown in Figure 9. (5) e- 15 /......,-i..., I 2 1..c:: n 1./ j" i //i./ Figure 7: Bode plot of P(s). -o -g ~= -loo._ -. "~ g. -15 t-- n I Note that the poles cancel between Gl(S) and G2(s ) when P(s) is fored. Also note that front steering inputs doinate at very high frequencies due to the difference in the relative order of the input transfer functions in Equation (3). The copensator Q(s) ust be chosen with soe caution. Since torque inputs are only desired to assist during rapid aneuvers, the loop gain P(s)Q(s) ust be sall at high frequencies and large at low frequencies. One ethod of achieving this would be to introduce a single very low frequency pole into P(s). However, this additional pole would introduce both a low frequency pole and ero into Gs~so(S). Thus, a higher-order controller Q(s) is required i... i...,... t..., Figure 9: Bode plot of Gslso(S). A controller C(s) is then designed using classical controller design techniques to achieve sufficient phase via a doublelead copensator. Co(s)_ 114.s s+424 (6) s s+3232 The resulting C(s)Gsiso(S) is shown in Figure 1. 1

5 2, _,.& Phase Margin =. Gain Margin = "~ 1 ~ ~ 1 ~ Figure 1: Bode plot of C(s)Gslso(S ). 5. Ipleentation on the Illinois Roadway Siulator To test the PQ controller of Section 4, both siulation and experiental tests were perfored using the Illinois Roadway Siulator (IRS) and a scale vehicle with front, rear, and torque-steering capability. The IRS is an experiental test bed consisting of approxiately 1/8 scale vehicles running on a siulated road surface, where the vehicles are held fixed with respect to inertial space and the road surface oves relative to the vehicle. An analogy would be wind tunnel testing of aerospace systes. There are any advantages to using scale vehicles for testing including cost, durability, repeatability, safety, and flexibility of the experient. Additionally, extensive testing has established a very high degree of dynaic siilitude between the scale test vehicles and full-sied vehicles [ 13, 14]. The vehicle's velocity of 4. /s was chosen to represent an average full-sied passenger vehicle operating at highway speeds of 65-7 ph. Further inforation about the treadill syste, vehicle setup, and dynaic siilitude can be found in [ 14].?iiii!;iiil i!:!i :iii:i iii i Figure 11" IRS 'Uberquad' Vehicle. The vehicle steering and torque inputs are actuated via peranent agnet DC otors. Because the otor torque is proportional to input current for DC otors, the wheel and steering torque can be coanded directly. Although this direct ethod torque input is not currently practiced on conventional vehicles, future vehicle designs using hybrid electric/internal-cobustion engine drive systes will likely have this capability. Further, the high-bandwidth drive otors on IRS scale vehicles can be ade to eulate a conventional drive/braking syste by odifying the control algorith to exhibit powertrain and braking dynaics. For this study, actuator dynaics were included in the test vehicle but not in the design steps. Using published values for full-sied vehicles as a guide, full-sied steering actuators can be approxiated as second order systes ranging in bandwidth between 3 and 6 H. All steering actuators on the vehicle were chosen to represent linear 5 H second-order actuators with critical daping. Since the dynaics are not included in the controller design, experiental testing on the vehicle containing such dynaics provides a easure of robustness of the resulting controller. The position of the center of gravity of vehicle was onitored via a light-weight ar ounted to the vehicle fro a fixed ground position. Monitoring the angles of the ar via integrated encoders allowed the vehicle states to be easured [ 14]..1 -ff ~-" -g._i Experiental Vehicle ~ ~... Experiental Vehicle ~. Siulation / ~O 5 1 o 1 t it o ~i lj Siulation!i \ Experiental Vehicle i ;, ; 1 Figure 12: Experiental results Shown in Figure 12 are the results of ipleenting the coordinated controller on the IRS vehicle syste. Clear frequency separation of the inputs is achieved while aintaining a siilar phase on the control inputs. Note that the vehicle response contains biases due to the lack of an integrator in the syste and the fact that the wheels are not exactly aligned. Additionally, it should be noted that the response is less daped than predicted priarily due to the presence of high-frequency actuator dynaics that i 11

6 introduce a control input delay. This delay is especially noticeable on the front steering input. To ensure that the high-frequency torque input was actually affecting the vehicle, the torque inputs were shut off in siulation with the result being vehicle instability. This clearly justified the use of torque inputs. Lateral position control at high speeds is known to be a difficult proble using only the lateral position state for feedback [15]. Our difficulty in achieving a high syste daping is consistent with this previous published work. With this in ind, the tracking results of Figure 12 are fairly good. Efforts were ade to increase the syste daping, but such efforts required higher-order controllers containing derivative approxiations that were unrealiable. In the IRS vehicle syste with encoder-quantied feedback, derivatives of higher order than 2 contain too uch noise for useful control purposes. Additionally, it was found that the pole assignent in designing Q(s) to perfor frequency-based control input weighting introduced very lightly daped eros close to the origin. These eros effectively liited the closed loop phase argin to 55 degrees. Usual ipleentations of the PQ controller on systes that naturally contain larger frequency separations usually involve systes whose subsyste pole locations do not cancel in the PQ design step. In such a case, the uncancelled poles assist in the frequency shaping of the control input weights. In the vehicle ipleentation, each branch of the MISO vehicle syste share the sae pole locations, therefore artificial poles ust be introduced that in later design steps becoe lightly daped eros. Clearly, a ore coputer-orientated approach such as an H-infinity ethod could be used for the choice of Q(s). However, here we present the basic concept and leave it for future work to explore the optial design of the controllers 6. Conclusions A coordinated MISO controller was introduced to achieve the high-speed lateral position-tracking task. The design ethod iposed both frequency separation constraints on the control inputs in addition to addressing traditional stability concerns. Experiental results on a scaled test vehicle confired the design. The results of both the controller design and the experiental ipleentation indicate a distinct benefit associated with the separation of input frequencies. This is not unlike the use of frequency weighting in other MISO control schees such as LQ regulation. The ain difference would be in the output feedback representation of the PQ versus a state feedback frequency-shaped LQR representation. However, the conceptual siilarities would be interesting topics for future investigations. REFERENCES [1] N. Matsuoto and M. 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