Quarter-Car Experimental Study for Series Active Variable Geometry Suspension

In this paper, the recently introduced series active variable geometry suspension (SAVGS) for road vehicles is experimentally studied. A realistic quarter-car test rig equipped with double-wishbone suspension is designed and built to mimic an actual grand tourer real axle, with a single-link variant of the SAVGS and a road excitation mechanism implemented. A linear equivalent modeling method is adopted to synthesize an H-infinity control scheme for the SAVGS, with the geometric nonlinearity compensated. Simulations with a theoretical nonlinear quarter-car indicate the SAVGS potential to enhance suspension performance, in terms of ride comfort and road holding. Practical features in the test rig are further considered and included in the nonlinear model to compensate the difference between the theoretical and testing behaviors. Experiments with a sinusoidal road, a smoothed bump and hole, and a random road are performed to evaluate the SAVGS practical feasibility and performance improvement, the accuracy of the model, and the robustness of the control schemes. Compared with the conventional passive suspension, ride comfort improvements of up to 41% without any deterioration of the suspension deflection are demonstrated, while the SAVGS actuator power is kept very low, at levels below 500 W.


I. INTRODUCTION
EHICLE suspension is a system of springs, shock absorbers and arms that connects a car body to its wheels and allows relative motion between the two [1]. Suspensions are widely studied to improve ride comfort and road holding. According to the nature of the force-producing elements, suspension designs are classified into three main types [2]- [3]: 1) Passive suspension: the main component is a conventional spring-damper unit, which passively adapts to the road profile and dissipates energy from road perturbations. This type of suspension still dominates the market due to low cost and easy maintenance; 2) Semi-active suspension: the structure is similar to a passive suspension but the damping force is actively adjusted by electro-rheological, or magneto-rheological Manuscprit  dampers, or solenoid valve controlled dampers; 3) Parallel active suspension: a separate (hydraulic, hydro-pneumatic, or electro-magnetic) actuator is employed in parallel with a passive spring or spring-damper unit to exert an independent force and provide improved performances. The performance of active suspensions can be further improved and realize its full potential by integration of advanced sensors and intelligent controls, such as road preview control [4]. More recently, a series active variable geometry suspension (SAVGS) has been proposed and theoretically proven as a valid alternative to existing solutions [5]- [12]. Fig. 1 illustrates the SAVGS application to a quarter-car with a double wishbone suspension. In addition to a conventional suspension assembly, a single-link arm is introduced between the chassis (the point 'G') and the upper end eye of the spring damper unit (point 'F'). The single-link is driven by a rotary actuator to actively improve the vehicle dynamics. The SAVGS features: 1) performance improvement in ride comfort and road holding (with additional improvements available in the full car case), 2) limited power demand in the single-link actuator, 3) small sprung mass increment and negligible unsprung mass increment, and 4) inherent fail-safe behavior [6]. Fig. 1. The Series Active Variable Geometry Suspension application to a quarter-car with double-wishbone suspension. 'RC' is the vehicle roll center and 'ICR' is the instantaneous center of rotation. θSL is the single-link angle with respect to the horizontal plane. The superscript (se) denotes the static equilibrium state with zero torque (TSL) applied on the single-link [6]. V An experimental study of the SAVGS with a quarter-car test rig is performed in this paper, with the main contributions being: 1) mechatronic system development of the SAVGS quarter-car test rig, 2) H-infinity control with geometric nonlinearity compensated and sensor noise penalized, for operating the SAVGS; 3) experimental validation of the SAVGS practical feasibility, the modeling accuracy, and the robustness of the control algorithms.
A variety of quarter-car test rigs have been set up for active suspensions study [13]- [22], as experiments are highly important to evaluate the practical implementation feasibility and control system robustness. However, two primary limitations often exist in these test rigs. Firstly, both the sprung and unsprung masses perform pure vertical motions, without accommodating the suspension kinematics (e.g. the installation ratio [23] and the lateral tire force). Secondly, parameters of realistic quarter-cars (e.g. the spring stiffness and the sprung mass) are scaled down, and therefore the actual practical feasibility cannot be accurately evaluated. The test rig built in this paper aims to address both of these issues.
The remainder of this paper is organized as follows. Section II presents the mechanical design and implementation of a SAVGS quarter-car test rig, including the design of the singlelink and road excitation mechanisms. Section III describes the control systems involved, in terms of both theoretical design and physical implementation. This includes: an adoption of a linear equivalent modeling method to synthesize an (outerloop) H-infinity control scheme for the single-link operation, the design of (inner-loop) actuator control, and the embedded system development in the rig for the overall control implementation. Section IV analyzes the SAVGS test rig by simulations with a theoretical nonlinear quarter-car model, and further develops this model to account for practical features of the test rig. Section V conducts quarter-car experiments over a sinusoidal road, a smoothed bump and hole, and a random road, and analyzes the results. Conclusions are drawn in Section VI.

II. QUARTER-CAR TEST RIG DEVELOPMENT
This section presents the mechanical design and implementation of the SAVGS quarter-car test rig. The baseline rig is built from an actual conventional passive doublewishbone suspension arrangement connecting an actual wheel (unsprung mass) to a set of steel plates (sprung mass) supported on a frame structure with a rail and carriage guiding system that allows only vertical motion of the plates. The active single-link with its actuation and transmission mechanism is then integrated, as well as a horizontal plate with a vertical excitation mechanism, which emulate road disturbances on the wheel. The suspension response is measured by related sensors. The mass properties and the spring-damper characteristics are in accordance with those in one quarter of a grand tourer (GT). The mechanical parameters of the GT quarter car, adopted by the rig, are listed in Table I in the Appendix.
The isometric view of the SAVGS quarter-car test rig is shown in Fig. 2. The development procedure comprises three stages: 1) the dimensioning of the suspension passive geometry, 2) the integration of the active single-link mechanism, and 3) the design of the road excitation mechanism.

A. Dimensioning of Passive Suspension
The double-wishbone suspension assembly utilized is from the rear axle of a Ferrari F430. It mainly comprises: one springdamper unit (Öhilins TTX40), one set of double-wishbone arms, one hub carrier and its attached road wheel. This assembly is constructed in the rig according to its conventional geometric arrangement, and therefore it can accurately emulate the actual suspension operation in a GT. The geometric parameters are listed in Table II in the Appendix, where the position of the roll center [23] is governed by the customized car body size with rolling and pitching stability taken into account, the static wheel camber angle is determined by the desired lateral force, and the installation ratio is mainly decided by the desired suspension stiffness and required suspension travel. These parameter are taken from Ferrari car datasheets, while the position of points 'A', 'C' and 'G' (see Fig. 1) in the passive case is to be determined.
Firstly, the cylindrical joints connecting the wishbones and the sprung mass (points 'A' and 'C') are dimensioned as follows: 1) With the given roll center height (lrc), half-track width (lht) and initial wheel camber angle (θIHy), points 'B', 'D', 'I', 'H' at their static equilibrium can be fixed; 2) Due to the fact that the wheel motion is constrained by both the upper and the lower wishbones, lines of 'A-B', 'C-D' and 'RC-H' intersect at the point 'ICR' (commonly referred to as instantaneous center of rotation) [23]. With this assumption, the position of 'A' and 'C' can be determined by an appropriate value of lICR-H (the distance between point 'H' and 'ICR'); 3) The suspension geometric arrangement also aims to minimize the changes in both the camber angle (θIHy) and the scrub displacement (∆yH) throughout the possible wheel travel (∆zH), so that the tire attitude can maintain the desired lateral handling ability (e.g. cornering action). By sweeping the value of lICR-H, the position of the points 'A' and 'C' can be determined with the minimal camber and scrub variation obtained. The corresponding wheel motion with the optimized lICR-H (= 2.92 m) is shown in Fig. 3, in which the camber angle (θIHy) and the scrub displacement (∆yH) vary only slightly within [-2.07˚, -0.86˚] and [-8.7 mm, +6.8 mm] respectively.  The installation ratio at the static equilibrium (se) is given by: and its value is taken from the vehicle manufacturer datasheet (RSD = 0.56). By applying the principle of virtual work to the unsprung mass (mu) at its static equilibrium configuration, we obtain: The equilibrium spring force () se SD F and vertical tire force () se tz F can be further written as: ) . horizontal plane) can be determined, and therefore the location of point 'G' is found. This subsection fully defines the geometric arrangement of the double-wishbone suspension in the passive case. The changes in the spring force (∆FSD) and the vertical tire force (∆Ftz) with respect to the wheel travel are illustrated in Fig. 4.

B. Design of Single-Link Mechanism
The single-link mechanism is introduced with the suspension geometric arrangement (at the static equilibrium) unchanged.
The single-link length of 11 mm has been selected for the rear axle of the GT [6], with the desired tire force increment and the actuator torque-speed envelope considered. The single-link can be stabilized at different angles by an equilibrium torque TSL from the single-link actuator. This torque is calculated by applying the virtual work principle to the single-link: With the sprung mass position assumed to be fixed, the single-link influence on the vertical tire force and the spring force is shown in Fig. 5. It can be seen that this SAVGS configuration is reasonable when compared with the results in Fig. 4, as the single-link rotation has the potential to compensate the tire force changes (ΔFtz) caused by the wheel travel propagation. Moreover, the SAVGS with the single-link at an offset angle (ΔθSL) of 90˚ (referred to as the nominal state) has more effective performance influence, as both the spring force (ΔFSD) and vertical tire force increment (ΔFtz) against the single-link angle (ΔθSL) have the maximum slope. The essential steady-state load analysis of the SAVGS and the calculation of the steady-state TSL also help to guide the selection of the singlelink actuator, as well as the following structural design. Fig. 5. Changes in the spring force (top), the vertical tire force (middle) and the single-link torque (bottom) with respect to the single-link angle. The sprung mass is assumed to be fixed. ΔθSL is the single-link angle increment with respect to its static equilibrium state. Fig. 6 illustrates the mechanical arrangement of the singlelink and its transmission assembly. Inspired by the "engine crankshaft", the single-link is integrated with the transmission shaft (accommodated by two bearings) via a "crankpin" with an offset of lSL = 11 mm. This "crankpin" acts as point 'F' (see in Fig. 1), which is aligned with the upper end eye of the springdamper unit. The two supporting bearings are attached onto the sprung mass, and they protect the gearbox output shaft from bending moments, with the axial torque (TSL) applied only. The single-link implemented in this way has benefits of structural simplicity and high torsional stiffness [24].

C. Design of Excitation Mechanism
An excitation mechanism is added to enable road disturbance forcing on the wheel. Due to space constraints in the test rig, a cam-driven excitation device is developed to convert the rotary motion of a cam into translational motion of a road plate. As shown in Fig. 7, an eccentric cylinder, which is driven by a rotary actuator, acts as the cam. A roller follower, which is always in contact with the cam, is fixed underneath the road plate. This arrangement forms a rolling friction pair between the actuation and the load, and it facilitates heavy load operation. Additionally, four linear bearings, together with linear rails, are mounted at each corner to guarantee pure vertical translational motion of the road plate. The road height (zr) variation against the cam angle can be derived through the geometric relationship between the cam and the roller follower, as shown by the solid line in Fig. 8. It is approximated by a sinusoidal function: where hCAM is the difference between the cam's tip radius and base radius. The cam, together with its transmission and actuation assembly, are accommodated inside two parallel H-profile beams (as shown in Fig. 9). Due to this space constraint, a cam with hCAM = 27.5 mm is chosen. Inevitably this limits the maximum peak-to-peak road height that is reachable in the experimental study to 27.5 mm, which is deemed to be adequate for the present purposes.

D. Test Rig Implementation
With the critical mechanical parts discussed above, the SAVGS quarter-car test rig is set up, as shown in Fig. 9, Hprofile beams constitute the supporting frame, making the whole test rig portable. The single-link at the static and nominal equilibrium is illustrated respectively in Fig. 9-a) and b). The cam and the roller follower in the road excitation mechanism are shown in Fig. 9-c).
In order to measure the system performance and to implement close-loop control: 1) a uniaxial accelerometer is attached on the sprung mass to measure its vertical acceleration, 2) a linear potentiometer is mounted between the sprung mass and the spring-damper lower end eye to obtain the suspension deflection, as shown in Fig. 9-d), 3) two PMSMs (permanentmagnet synchronous motors) are selected to drive the singlelink and the cam separately, with the actual position, velocity and torque acquired by internal quadrature optical encoders and current sensors, and 4) a real-time controller is adopted to communicate with an industrial computer (for the humanmachine interaction), two PMSMs, and all the sensors mentioned above.
In summary, the features of the SAVGS quarter-car test rig are: 1) all the mass properties are the same as an actual GT (Ferrari F430 real axle), 2) the suspension kinematics (e.g. the installation ratio) is considered, as an actual suspension assembly is utilized and reconstructed, 3) the target road profiles can be accurately generated through the motion control of the cam actuator, 4) real-time control and monitoring of the embedded system is employed, and 5) the test rig system enjoys structural simplicity and compactness. Fig. 9. Mechanical design and implementation of the SAVGS quarter-car test rig. It mainly comprises: 1) a cam-driven excitation mechanism, 2) a doublewishbone suspension assembly, 3) a single-link and its actuation assembly, and 4) a sprung mass assembly with vertical translation only.

III. CONTROL DESIGN AND IMPLEMENTATION
This section specifies the SAVGS overall system control design and implementation, as shown in Fig. 10. Firstly, an outer-loop H-infinity control is synthesized to enhance both the ride comfort and the road holding, with an equivalent linear model utilized. Secondly, an inner-loop actuator control is also developed to track the desired single-link velocity (ωSL*), whilst ensuring that none of the actuator constraints (voltage, current, power, torque and speed) are violated. Finally, a mechatronic system of the SAVGS test rig is constructed for the overall control implementation, including the H-infinity control, the actuator speed tracking, ωSL* and ωCAM*, in the single-link and excitation mechanism respectively. Fig. 10. Schematic of the SAVGS test rig system overall control. y corresponds to the system measurements, while e to the suspension performance objectives. ωSL denotes the angular velocity of the sing-link while ωCAM denotes that of the cam. żr is the vertical road velocity.

A. Linear Equivalent Model
A linear model of the quarter-car test rig is necessary for Hinfinity control synthesis. In previous studies of active suspensions, linear models with two pure vertical motions are widely applied, in which the geometric nonlinearity is not considered. In [12] a nonlinear model is firstly built and then linearized at a trim state of ΔθSL = 90˚ to synthesize an H-infinity control for the SAVGS quarter-car. However, the linear model obtained in this way is not accurate when a large single-link offset is required.
An equivalent modeling method, proposed in [10]- [11], is adopted for the SAVGS quarter-car test rig, as shown in Fig.  11. The equivalent model is purely linear, with the geometric nonlinearities associated with the single-link angle variation (ΔθSL) lumped into the function α. The following assumptions are made to derive the linear equivalent model from a nonlinear multi-body model of the system [24]. 1) Geometric equivalence: both models have the same suspension deflection (ls = zuzs, the vertical distance between the sprung mass center and the unsprung mass center) and the same tire deflection (lt = zrzu, the vertical distance between the unsprung mass center and the road surface). The suspension deflection in the equivalent model is: where Δzlin is the displacement increment of the linear actuator with respect to its nominal equilibrium state: and () eq SD l is the equivalent spring-damper length. 2) Actuator equivalence: both the actual rotary actuator and the equivalent linear actuator must have the same power output: where () eq SD F denotes the equivalent spring force. 3) Spring-damper equivalence: the equivalent spring needs to satisfy Hooke's law, while the equivalent damper has the same energy dissipation as the actual one: where lSD is the actual spring-damper length (the distance between points 'E' and 'F'). Coefficient functions, which are for the conversion from the multi-body model to the equivalent one, are listed in (10). They are all dependent on the single-link angle (ΔθSL) and the lowerwishbone angle (ΔθLW): Equation (10) can be derived from (6)- (9). The variation of each coefficient against ΔθSL and ΔθLW is shown in Fig. 12 to Fig. 14. Both keq and ceq vary only marginally with ΔθSL and ΔθLW, and are therefore approximated as constants by their value at the nominal state (γk = 0.3806, γc = 0.3604), in which the single-link offset is 90˚, shown as the red marks in Fig. 12 and Fig. 13 respectively. The dependence of function α with the lower-wishbone angle (ΔθLW) is small, therefore it can be approximated by a parabolic equation with respect to the singlelink angle (ΔθSL) only, as shown by the dashed line in Fig. 14.
The motion equations of the SAVGS linear equivalent model are listed in (11), with the mass of the equivalent linear actuator neglected due to the small inertia of the single-link:

B. H-infinity Control Synthesis
The H-infinity methodology is widely applied to linear MIMO (multi-input multi-output) systems due to its robustness. This control technique aims to minimize the disturbance propagation to the system performance objectives. The standard configuration of the H-infinity synthesis is illustrated in Fig. 15.  Fig. 15. Generalized regulator configuration of the H-infinity control application. P corresponds to a linear plant in state-space representation, K to the synthesized H-infinity controller, d to the exogenous disturbance, u to the manipulated variables, e to the performance objectives to be minimized, and y to the system measurements for feedback [27].
The H-infinity controller designed for the SAVGS quartercar in [10]- [11] is modified and adapted to the rig, as follows: 1) The exogenous disturbance in the rig is: d = [d1 d2]. d1 is an exogenous command signal of the linear equivalent actuator displacement, Δzlin(e), to enable zero or low frequencies (0-1 Hz) tracking, for example to set the nominal position of the singlelink (ΔθSL = 90˚). d2 is the vertical road velocity. The SAVGS has been theoretically proven to be effective with both disturbance of the vertical road velocity and the load transfer [10]. However, the latter is not included here, as it cannot be emulated by this quarter-car test rig.
2) The performance objectives to be minimized are e = [e1 e2 e3 e4]. e1 is the Δzlin(e) tracking error. e2 and e3 are the penalized sprung mass acceleration and tire deflection respectively, both of which are the major objectives to evaluate ride comfort and road holding in a quarter-car. e4 is the control effort, which is penalized to confine the single-link operation within a specified working frequency range (0-10 Hz), and also to reject the sensor noise to eliminate the output ripple of the actuator.
3) The measurements y = [y1 y2 y3] are selected given the feasibility of the sensors: y1 is the suspension deflection measured by a linear potentiometer. y2 is the vertical acceleration of the sprung mass, which is acquired through a uniaxial accelerometer. y3 is the tracking error of the equivalent linear actuator displacement, which is obtained by an optical quadrature encoder in the single-link actuator.
4) The manipulated variable u is the command velocity of the linear equivalent actuator (żlin).
Weighting functions penalize the importance between multiple objectives at different frequency ranges, and they are also tuned to reject sensors noise, as follows: 2) In the case of the weighting functions of the performance objectives [Werr, WAcc, WTD, Weff], their cornering frequency is selected to match the desired frequency bandwidth of the active suspension, while their DC values can be tuned to be either conservative or aggressive. A second-order transfer function Weff is applied to also reject the accelerometer noise (concentrates between 10-20 Hz), which results in the output torque ripple of the actuator and the single-link vibration.
A first-order transfer function (with the cut-off frequency assumed to be 17.8 Hz [11]) is adopted to approximate the limited bandwidth of the actuator frequency response behavior.
With the identified input and output variables and the tuned weighting functions, the configuration scheme for H-infinity control synthesis is shown in Fig. 16. The "Quarter-car (eq)" is the linear equivalent model in statespace representation. The constructed plant (top dashed block in Fig. 16) corresponds to 'P' in Fig. 15. By means of the MATLAB command hinfsyn, an H-infinity controller 'K' is synthesized. The transfer function 'M' filtering the output y3 has a free integrator and aims to ensure a zero steady-state tracking error of the exogenous command (Δzlin(e)) [11], [27]. In order to facilitate simple tuning of the desired tracking response characteristics in the time domain (e.g. rise time) by experiments, 'M' is added and tuned a posteriori, and it is not part of the H-infinity synthesis. The overall controller (bottom dashed block in Fig. 16) includes both the synthesized 'K' and the experimentally tuned 'M', which is given by:   Theoretical analysis with the linear equivalent model is performed: bode diagrams of the overall controller without 'M' and with 'M' are shown in Fig. 17. It can be seen that the sole controller 'K' is a band-pass filter for all inputs. The path from 'y3' to 'u' is changed to a low-pass filter due to block 'M', as it takes effect at low frequencies to zero the Δzlin(e) tracking error.
Bode magnitude diagrams of the linear equivalent model in passive (i.e. Δzlin is fixed at the value of 0) and active cases are plotted in Fig. 18 and Fig. 19. The synthesized controller significantly improves the suspension performance, as the objective gains are attenuated (by 6-8 dB) at around the humansensitive frequency of 2 Hz. The addition of 'M' slightly affects the frequency response of the sprung mass acceleration and tire deflection, and it only increases the suspension deflection gain below 1 Hz, as the actuator takes effect to track the linear actuator reference displacement (Δzlin(e)) at low frequencies (also shown in Fig. 19).
Overall, the addition of the transfer function 'M' benefits the zero tracking error of the exogenous command, without affecting the close-loop system performance. Therefore 'M' included in the final design.
The linear analysis in this section gives an indication of the expected SAVGS performance improvement over the passive suspension. Further and more accurate demonstration of the dynamic performance improvement is provided later by nonlinear simulations and experiments in Sections IV and V.

C. Single-Link Actuator Modeling and Control
A permanent-magnet synchronous motor (PMSM, also known as brushless AC motor) is selected as the single-link actuator due to its high power density, high efficiency and fast response. The classical "d-q" vector modeling and control algorithm (also referred to as field-oriented control) is commonly adopted in AC motors [28]- [29]. The basic idea is to transform three-phase stator current into two independent variables: a magnetic field-generating part (id) and a torquegenerating part (iq). The electrical dynamics of this equivalent d-q model are written as: where Ls and Rs are the d-q phase resistance and inductance, vd and vq are the voltages in the d-axis and q-axis respectively. ψd = Lsid + ψa and ψq = Lsiq are the flux linkages in d-q reference frames, ψa is the flux linkage of the rotor permanent magnet, and ωe is the electrical frequency. The output electromagnetic torque (Tem) of the PMSM is then given by: where npp is the number of the pole pairs. The PMSM torque control is based on the widely utilized "zero d-axis current control" strategy, in which id is forced to be zero to ensure maximum torque-per-amp operation [30]. Equation (16) Fig. 20. Cascaded control scheme for the single-link actuator [32]. Ggbx (= 40) is the single-link gearbox ratio. ωhss is the high-speed shaft velocity of the actuator, while ωSL is the velocity of the single-link, which is connected to the low-speed shaft of the actuator. Gaff is the feed-forward gain in the velocity control loop.
As both npp and ψa are constant values for a specific PMSM, the actuator torque, Tem, is directly proportional with the q-axis current iq. Therefore, the PMSM actuator can be simplified as a DC equivalent motor [30].
A cascaded velocity-current control scheme is performed for the single-link motion control, as shown in Fig. 20. The actuator velocity ωSL are acquired through an optical quadrature encoder, and the actual current signals id and iq are obtained by transforming measurements of the phase currents ia, ib and ic.
In the velocity loop, an acceleration feed-forward path is added in parallel with a proportional-integral controller to contribute to the reference signal tracking without further increasing the proportional gain. Moreover, velocity and acceleration/deceleration clamps are in place for safety. A first order low-pass filter is applied to reduce the torque ripple and resonances from the motor and gearbox.
In the current loop, according to the "zero d-axis current control" strategy, the reference d-axis current id* is forced to zero, while a proportional-integral controller is applied to the iq* tracking. The output voltages vd* and vd* are converted into three-phase AC voltage va*, vb* and vc* (by means of the d-q inverse transformation) before feeding the inverter in the PMSM servo drive. Additionally, a current clamp (0/+7.5 A) is added to limit the magnitude of the actuator high-speed shaft output torque and also to prevent the same torque from ever becoming negative thus avoiding the gearbox backlash effect. The power of the single-link actuator is intentionally limited to 500 W, which is a reasonable value for the SAVGS full-car application in terms of the actuator size and weight.
The main parameters of the single-link actuator and of the tuned controllers are listed in Table III in the Appendix. Another PMSM (with a rated power of 3610 W and a rated torque of 9.85 N·m) is selected to drive the excitation mechanism with the same cascaded control scheme applied.

D. Control Implementation in Nonlinear System
The H-infinity control and the single-link actuator control are presented in subsections B and C respectively, and they are further developed to adapt to the nonlinear mechatronic system of the SAVGS quarter-car test rig, as shown in Fig. 21. 1) The H-infinity control provides the reference velocity of the equivalent linear actuator (żlin*), which is converted into the single-link reference velocity (ωSL*) by function α = żlin/ωSL. The actual single-link position (∆θSL) is converted into the displacement of the equivalent linear actuator (∆zlin) by function β = ∆zlin(∆θSL), for utilization by the H-infinity controller.
Functions α and β are plotted in Fig. 22, where α is the parabolic approximation shown in Fig. 14 and β is the integral of α with respect to the single-link angle (ΔθSL).
2) The exogenous command of the equivalent linear actuator displacement (Δzlin(e)) is fixed at the value of its nominal equilibrium state (corresponding to ΔθSL = 90˚), where the single-link has the maximum control influence [11].
3) The controllers are designed in continuous time but are subsequently discretized, due to the fact that the test rig physical controller, NI cRIO-9022 [33], has 200 Hz I/O sampling frequency. The discretization is implemented by MATLAB command c2d (sys, 0.005, 'zoh'), 5 ms is the sampling period.    Subsections A-D theoretically complete the overall control design and implementation of the SAVGS quarter-car test rig, while the practical application is based on the test rig mechatronic system, which is constructed as follows.

E. Test Rig Mechatronic System
The electrical configuration and the software deployment are illustrated in Fig. 23. A physical controller (NI cRIO-9022) is acting as the core in this system. It is equipped with a real-time micro-processor, a FPGA module, I/O modules and Ethernet modules [33]. This controller coordinates the electrical hardware of: 1) a servo drive (Kollmorgen S300) for the singlelink actuator, 2) a servo drive for the excitation actuator (Kollmorgen AKD EtherCAT), 3) sensors of a uniaxial accelerometer and a linear potentiometer, and 4) a PC.
The single-link actuator and its servo drive are utilized to convert the command velocity (ωSL*, from an analog output of the physical controller and fed into an analog input of the servo drive) to the actual one (ωSL). As shown in Fig. 23 (middle dashed block), this actuator is a PMSM powered and controlled as follows. Firstly, a rectifier converts the AC mains to a DC voltage (= 320 V), and it feeds the DC bus in a PWM inverter. Secondly, the Space Vector Modulation (SVM) technique is adopted in the inverter to generate three phase sinusoidal voltage (va, vb, vc) for the AC machine, with their reference signal (va*, vb*, vc*) given by the torque loop in the actuator controllers. It is worth noting that the AC mains power supply and the rectifier are only relevant to the test rig, whereas a DC battery and a low to high voltage DC-DC converter are to be employed on a real car instead [8].
The excitation actuator is also a PMSM with a similar working principle as the one for the single-link. As shown in the bottom dashed block of Fig. 23, this actuator aims to convert the road profile reference signal (ωCAM*, θCAM*) to the actual cam angle/velocity (ωCAM, θCAM). By using an advanced EtherCAT servo drive, this excitation actuator can communicate with the physical controller (NI cRIO-9022) through the EtherCAT protocol. In this way both of the input (ωCAM*, θCAM*) and output (TCAM, ωCAM, θCAM) signals can be digitally transmitted with reduced signal noise.
LabVIEW software is utilized to develop an embedded control and monitoring platform.
LabVIEW Program #1 is deployed in the real-time microprocessor (NI cRIO-9022), where critical control algorithms and communication protocols are mainly implemented, including: 1) the discrete H-infinity control, 2) the velocity loop control for the excitation actuator, 3) the communication with the FPGA module (access to I/O modules for data acquisition), and 4) the communication with a PC by means of network protocols. LabVIEW Program #2 is deployed in the industrial PC for the human-machine interaction. This program includes: 1) command handlers (e.g. motor initialization/stop and passive/active switch), 2) waveform display (for the monitoring of all variables of interest), 3) data logging (e.g. the sprung mass acceleration and the suspension deflection), and 4) further signal analysis based on historical records.
Compared to LabVIEW Program #1, tasks in Program #2 have lower priority and are less required in real-time. However, functions like data logging and waveform display consume larger memory space. Therefore, these tasks are split from the real-time processor and deployed in the PC. The benefits in this way are that: 1) the memory in the real-time micro-controller is released, and 2) the deterministic loops and iterations (with a fixed execution period of 0.005 s) in the critical controllers (the outer-loop H-infinity control and the inner-loop velocity tracking) are guaranteed.

IV. NONLINEAR SIMULATION AND ANALYSIS
With the mechanical structure, the electrical configuration and the control strategies discussed in Sections II and III, simulations with the nonlinear SAVGS system are performed in this section to theoretically analyze the test rig behavior.
A theoretical nonlinear quarter-car model with the actual rig parameters (referred to as "SimQC") is firstly built to explore the SAVGS potential in enhancing the suspension performance. In order to bridge the gap between the simulation results of the theoretical model and the experimental results of the actual test rig, practical features (e.g. the railway-sprung mass friction) existing in the rig are further included to construct a more realistic model (referred to as "SimTR").

A. Nonlinear Modeling of Quarter-Car and Test Rig
The SimQC includes the following nonlinearities [6]: (i) the kinematics of the multi-body suspension assembly, (ii) the nonlinear damping force characteristics, and (iii) the electrical behavior (e.g. the saturations of the power, voltage, current, position, speed and torque) of the single-link actuator.
In addition to these nonlinearities, the SimTR model (as shown in Fig. 24) includes the practical features existing in the rig: (i) the sprung mass-rail way friction frail and the wheel tireroad plate friction Fty, (ii) the sensors noise in the potentiometer and the accelerometer measurements, and (iii) the system discreteness from 200 Hz sampling frequency in the controllers' I/O ports [33].
where ẏH is the lateral velocity of the tire contact point with the road plate (point 'H' in Fig. 1), ẏHth (= 0.05 m/s) is a shaping constant, and μt (= 0.25) is the friction coefficient between the tire and the Teflon road plate.
The following subsections (B to D) present nonlinear simulation results with three typical road profiles.

B. Simulation with Harmonic Road
The frequency at around 2 Hz is thought to be human sensitive [34]. Fig. 25 shows the time response over a sinusoidal road disturbance, with 2 Hz frequency and 2.75 cm peak-topeak amplitude. The comparison between "SimQC passive" and "SimQC active" shows that the SAVGS can significantly attenuate the performance objectives, with a 67%, 63% and 64% (RMS) drop in the sprung mass acceleration, the suspension deflection, and the vertical tire force respectively. On the other hand, the comparison between "SimQC passive" and "SimTR passive" shows the dynamics attenuation caused by the sprung massrailway friction (frail), while the robust controller is able to maintain "SimQC active" and "SimTR active" nearly identical.

C. Simulation with Smoothed Bump and Hole
A bump is a vertical road deflection commonly utilized to slow motor-vehicle traffic in order to improve safety conditions, while a hole is a structural failure.
Nonlinear simulation is performed over a smoothed bump and hole, both of which can be approximated by sinusoidal functions. The road height difference, 2.75 cm, is constrained by the profile of the utilized cam. The simulation emulates the case of a car travelling at a speed of 10 km/h and running over a smoothed bump/hole with 1.4 m width respectively at 0 and 2 s (the parameters selected here concern the rated torque-speed envelope of the excitation actuator). The results are shown in Fig. 26. The comparison between "SimQC passive" and "SimQC active" shows once again the performance improvement by the SAVGS: the sprung mass acceleration, the suspension deflection and the tire force drop by 47%, 21% and 42% (RMS) respectively for the bump case, and 47%, 23% and 43% (RMS) when the car experiences the hole. Further comparison between "SimQC passive" and "SimTR passive" also shows the dynamics attenuation by the rig railway friction. Despite the experimental uncertainties (e.g. the friction, the sensor noise, and the controller discreteness), "SimTR active" still matches "SimQC active" due to the controller robustness.

D. Simulation with Random Road
The random road is generated according to [35], where the road profile is described in the form of power spectral density (PSD):  (20) variable n is the spatial frequency (cycles per meter), n0 and ω are constants while k = 2-9 corresponds to road classes A-H, which reflect the roughness of the road surface (dashed parallel lines shown in Fig. 27 (21) where Ai is the amplitude corresponding to the spatial frequency ni, while ϕi is a random number distributed in the interval [0, 2π]. The emulated driving speed V and the spatial frequency range [nmin, nmax] are reconsidered with the following experimental limitations taken into account: 1) the sampling frequency of the physical controllers in the rig is 200 Hz [33], thus the maximum frequency in the generated road profile should be below 100 Hz, and 2) the position/velocity control loop in the excitation actuator has a bandwidth of 0-75 Hz [32]. Given these two limitations, the parameters of the random road profile for both nonlinear simulation and experimental study are selected as: the vehicle driving speed V = 60 km/h, the spatial frequency range [nmin, nmax] = [0.01, 1] cycles/m, the road length L = 1 km, and the spatial frequency step Δn = 0.001 cycles/m (Δn ≤ 1/L should be satisfied). Additionally, the actual test rig can only deal with class A due to the constraints of the cam radius and the rig space.
The PSD estimate of the modified Class A road profile for nonlinear simulations as well as experimental study is shown as the solid line in Fig. 27. The nonlinear simulation PSDs of the sprung mass acceleration and the vertical tire force are shown in Fig. 28. The comparison between "SimQC passive" and "SimQC active" shows the potential in the performance enhancement by the SAVGS, as the objective gains (the sprung mass vertical acceleration and the tire force) at around the human-sensitive frequency (1-3 Hz) are appreciably attenuated. The comparison between "SimQC" and "SimTR" indicates that the influence of the railway friction is overall slight, as it can either attenuate or intensify the sprung mass vibration with irregular road profiles.

V. EXPERIMENTS AND DISCUSSION
In this section, experiments with the SAVGS quarter-car test rig are performed to evaluate the practical feasibility (with the ride comfort mainly considered) and the accuracy of the modeling assumptions. The same road profiles discussed in Section IV are tested to compare directly with the nonlinear simulation results.

A. Tests with Harmonic Road
Test results over a 2 Hz harmonic road profile are shown in Fig. 29. The comparison between "Test passive" and "Test active" shows notable performance improvement of the SAVGS, as the sprung mass vertical acceleration and the suspension deflection decrease by 41% and 31% (RMS) respectively [24]. The comparison between "Test" and "SimTR" suggests the accuracy of the nonlinear models, as the response in both the passive and active cases are matched, despite the largest discrepancy in the suspension deflection. Fig. 29. Test results over a harmonic road profile, with 2 Hz frequency and 2.75 cm peak-to-peak amplitude [24]. The variables depicted are (top to bottom): sprung mass acceleration, suspension deflection, single-link angle, and single-link velocity.

B. Tests with Smoothed Bump and Hole
Test results over a smoothed bump and hole are shown in Fig.  30. The comparison between "Test passive" and "Test active" shows that the SAVGS can practically enhance the suspension performance, with 33% drop in the sprung mass vertical acceleration at the bump and 38% at the hole. The comparison between "Test" and "SimTR" shows that the model developed is accurate. However, the largest discrepancy occurs in the suspension deflection, which can settle at different levels (see for example the Δls for the "Test active" case in Fig. 30) as a result of the tire's lateral deformation, the tire-road plate static friction, and the sprung mass-railway static friction. By further observing the single-link angle (ΔθSL in Fig. 30), it can be seen that the SAVGS controller is capable of tracking the exogenous command of the equivalent linear actuator displacement (Δzlin(e), set at the value of its nominal state), as the single-link angle returns at 90˚ when the road disturbance ends.

C. Tests with Class A Random Road
The test results over the Class A random road (PSD estimate of which is shown as the solid red line in Fig. 27) are shown in Fig. 31. The comparison between "Test passive" and "Test active" shows that the SAVGS can improve the ride comfort even under a small road disturbance, as the sprung mass vertical acceleration is reduced at around human-sensitive frequency (1-3 Hz). The SAVGS performance marginally deteriorates at around 6-7 Hz, which can be tolerated especially with worse road profiles (e.g. Class B and C, with simulation results presented in Fig. 28).  The comparison between "Test" and "SimTR" shows that they have similar profiles in the frequency domain. Due to the fact that the Class A random road has a small road disturbance input, the influence of the experimental uncertainties on the sprung mass acceleration is relatively significant, including: 1) sensors and environment noise, 2) the complexity introduced by friction (e.g. the tire-road plate friction), and 3) the complexity of the tire's lateral deformation, and so on.

VI. CONCLUSION
A quarter-car experimental study of the series active variable geometry suspension (SAVGS) is performed in this paper. A test rig of a GT double wishbone suspension is designed, with an active single-link variant and a road excitation mechanism implemented. A linear equivalent modeling method is adopted to synthesize an H-infinity control scheme, with the geometric nonlinearity compensated. The designed controller actively adjusts the single-link velocity at higher frequencies to improve the suspension performance or it stabilizes the single-link position at low frequencies by tracking an exogenous command.
The SAVGS (outer-loop) H-infinity control as well as the (inner-loop) single-link actuator control are further modified and adapted to the experimental test rig application.
The simulation results with a theoretical nonlinear quartercar model indicates the potential of the SAVGS in improving both the ride comfort and the road holding. The practical features (e.g. the rig friction) existing in the rig are further considered and included in a nonlinear test rig model. This more realistic model compensates the testing behavior difference with the theoretical one, and also indicates the influence of the practical features on the experimental demonstration.
The experimental results with a sinusoidal road, a smoothed bump and hole, and a random road profile show that: 1) the SAVGS is practically feasible in both the mechanical implementation and the control application, as it significantly contributes to the ride comfort enhancement, 2) the SAVGS overall control is robust, as it is capable of dealing with the rig friction, the sensors noise and other experimental uncertainties, and 3) the mathematical assumptions of the model are essentially correct, despite the complexity in the tire deformation and the rig friction.
The demonstrated feasibility of the SAVGS to improve suspension performance and the development of suitable control strategies open new avenues for the exploitation of this solution in the automotive sector.