Target Audience: Vehicle dynamicists, Research scholars, FSAE and Baja students
Read Time: 15 min
Author: Aravind R Nair

Handling is one of the most discussed subjective analysis topic done for the vehicle dynamic estimation for a given vehicle, as it defines the vehicle’s competence, drivability, safety and in racing, has a good deal of impact in lap times. Being the determining parameter for stability and handling, it is vital that even from the earlier stages of a vehicle development, there should be a good theory for estimation and strong validation on the physical vehicle on that achievable performance.

Over the years, automotive engineers world-wide have come into accordance that this vehicle’s steer stability factor is dependent on changes in steering angle with respect to its lateral acceleration. This later differentiated the steering behaviors into “under” and “over”-steer properties, as described by Maurice Olley. According to Olley, The vehicle understeers if its path curves away from the side force, and oversteers if the path curves inward the side force direction.

The term Understeer gradient has been defined as a lateral acceleration dependent parameter that describes a vehicle’s handling property. It can be alternatively defined as the tendency of the vehicle to require further or lesser steering angle than a vehicle’s Ackermann steering angle.

It has been generally observed that vehicles with understeering behavior tends to be more responsive to steering inputs in transient disturbances as well. This implies the general trend that the stable daily drives today are generally made with a linearly understeer behavior, to increase safety and confidence to a wide range of drivers. This is due to the enhanced lateral and yaw damping in the tire’s linear range, good amount of directional stability and response time perceived by most drivers.

To test for the limit, a driver makes an input change while checking the vehicle’s response.  In order to test for the understeer limit in a skidpad, a driver would increase steering while paying attention to the vehicle’s movement.  If the front tires respond to increased steering by moving inward and tightening the vehicle’s line, then it was not yet at the understeer limit, as the front tires still had some capacity remaining.  If on the other hand, the vehicle did not tighten its line, then the front tires had already reached the limit and continuing past this will increase induced drag, causing it to slow down.  In order to avoid this, the driver would need to reduce steering until the front tires just start to move outward again, indicating they are once again below the limit.  At this point, the driver can test for the limit again by increasing steering once more.

If more throttle increases speed but the vehicle is able to maintain its current line, then it is not at the understeer limit yet.  Likewise, if increased speed forces the vehicle onto a wider line, then it was already at the understeer limit. A driver will typically use a combination of two inputs to test for the limit depending on which portion of a corner they are in with steering and throttle being used in combination during corner exit, and steering and brakes during corner entry.

With constant steering wheel movements to maintain skidpad path, the associated changes in vehicle movement are subtle and can be hard to see because the front tires are right at the limit. If the tires were below the limit however, those same back and forth steering motions would cause the car to move side to side noticeably.

The reason for this difference is the progressive nature of tire response.  This diagram shows how a typical tire has an initial linear increase in force, which then begins to taper off to the peak before steadily dropping.  The result of this is that a steering wheel movement done below the limit in the linear and early transitional area has a much greater effect on tire force and therefore vehicle movement then that same steering movement done near the limit.

Mainstream testing methods of Understeer gradient

The following list of testing methods have been adapted by the Automotive Industry for estimating steady-state, transient and dynamic Understeer gradient for their developments.

1. In SAE Standards J266, Four different tests are considered for determining understeer:

• Constant radius test: This is a closed loop test where the vehicle is driven in a constant radius turn. First, the radius is driven at the slowest speed possible to determine the Ackermann steering angle. Speed is increased in small increments and the vehicle is held in equilibrium to determine the required steering angle at each speed (and thus lateral acceleration) driven. Speed is increased until the vehicle is no longer able to maintain the radius. The data is typically presented in an X-Y graph with steering wheel angle on the vertical axis and calculated lateral acceleration (speed x speed / radius) on the horizontal axis.
• Constant Steer Angle Test: The constant steer angle test is an open loop test where the vehicle is driven at various speeds using a fixed steering angle. First, the vehicle is driven at very slow speed to determine Ackermann steering radius for the selected steering angle. Then vehicle is driven at increasing speeds steady state until the maximum lateral acceleration level of the vehicle is reached. The radius of turn at each speed is compared to the Ackermann radius to determine understeer.
• Constant Speed Variable Radius Test: This closed loop test is performed at many discreet radii at a constant speed. The vehicle is driven onto the circle in a steady straight line, then turned onto the circle. The portions of the test where the vehicle is at equilibrium are used to calculate the understeer gradient. The data is typically presented in an X-Y graph with steering wheel angle on the vertical axis and calculated lateral acceleration (speed x speed / radius) on the horizontal axis.
• Constant Speed Variable Steer Angle Test: This open loop test is driven at a constant speed for several seconds in a straight line, then for several seconds more after a selected steering input is made. The portions of this test where the vehicle is in equilibrium are used to determine understeer gradient.

2. Slowly Increasing Steer (SIS) test by NHTSA: In 2004, the NHTSA had conducted testing utilizing the Slowly Increasing Steer (SIS) protocol on two fifteen passenger vans. This maneuver is basically a constant speed, variable steer understeer test where the steering wheel angle is ramped up in a linear fashion. Testing was conducted at two loading conditions: Nominal Load (i.e. two occupants) and Maximum Occupancy (i.e. fifteen occupants).

The Slowly Increasing Steer maneuver provides data to assess the amount of turning capability of a vehicle (the Maximum Attainable Lateral Acceleration) and whether the vehicle’s handling degrades gracefully at the limit (did the vehicle plow or spin when the maximum achievable turn was attained). They have performed this maneuver for every vehicle tested during Phases II, III, and IV of NHTSA Rollover Research. Based on our experience we believe that this maneuver can be performed with excellent objectivity and repeatability.

3. Renfroe’s “Transient Oversteer Metrics”: All known definitions of understeer and oversteer used by vehicle manufacturers worldwide have related only to steady-state or quasi steady-state vehicle handling behavior. Recently, one organization has attempted to redefine the concept of oversteer and understeer as a property which can be considered in highly transient limit turn maneuvers.
   The “Transient Oversteer Metric” is calculated by first subtracting, at every point in time, the measured yaw rate in the test from the “Ackermann yaw rate”. Simply defined, the “Transient Oversteer Metric” is the rate of change over an arbitrary section in time of how much the actual measured yaw rate differs from the yaw rate predicted by a simple two degree of freedom bicycle model with tires sticking to the ground in a steady turn.
   However, does not have mainstream acceptance in the field and is highly dependent on driver inputs and not any vehicle handling characteristic.

How to evaluate the test outputs?

Now that it has been discussed on how to measure and quantify understeer gradient for various state conditions, there should be methods to be used to interpret the results using intuitive visualization and comparison. The following methods discuss the typical methods described in research:

1. MRA Moment Method (MMM): Described in “Race Car Vehicle Dynamics” by the Milliken brothers, MMM is a vehicle handling evaluation method developed between a contract research between General Motors and Milliken and his associates.
   The Yawing moment (Cn) of the vehicle can defined a test vehicle’s steering control, directional stability and yaw damping. The lateral force (Ay) determines the vehicle’s ability in holding its path during cornering. Both these parameters are functions of the Vehicle slip angle (beta) and the steer angle (delta) and hence, a plot between CN and AY are plotted after testing in the full range of delta and beta in both acceleration and braking.
   
2. “Four Parameter Lateral Transient Response” by Mimuro: This method uses a 2-DOF model of vehicles, and uses four parameters signifying yaw response: Steady state gain a1, natural frequency fn , system damping ratio delta and system phase delay phi on the individual arms of axes. This creates an overall rhombus, whose orientation gives an intuition on evaluating the handling ability and mode.

    

3. Massimo Guiggiani’s “Map of Achievable Performance (MAP)” approach: Described in his book “The Science of Vehicle Dynamics, this new “global approach” aims to close gaps between mainstream understeer evaluation methods to be used in all applications.
   Two plots are proposed for handling imaging, first one “curvature (rho) vs steer angle” and second “curvature (rho) vs vehicle slip angle (beta)”. The vehicle will be tested until it reaches four extremities: zero lateral acceleration, max speed, max lateral acceleration and max steer.

To summarize, for safety and performance quantification, it is vital to create a realistic estimate on the Understeer gradient, to ensure the imagined performance to be captured for the vehicle’s physical tests, so that the ground up design follows as the design intend. We have also discussed how different state considerations (Steady, transient and dynamic) should be measured and different approaches for understeer evaluation could be done.

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References:

1. Massimo Guiggiani. (2019). SCIENCE OF VEHICLE DYNAMICS: handling, braking, and ride of road and race cars.
2. Milliken, W. F., & Milliken, D. L. (1995). Race car vehicle dynamics. Sae International.
3. Topping, R. (2012). Understeer Concepts with Extensions to Four-Wheel Steer, Active Steer, and Time Transients. SAE International Journal of Passenger Cars – Mechanical Systems, 5(1), 167–186. https://doi.org/10.4271/2012-01-0245
4. Tandy, D. F., Colborn, J., Bae, J. C., Coleman, C., & Pascarella, R. (2015). The True Definition and Measurement of Oversteer and Understeer. SAE International Journal of Commercial Vehicles, 8(1), 160–181. https://doi.org/10.4271/2015-01-1592