## Kinematic Basics and the Control-arm IFS System

Arguably, the most common independent suspension design consists of a spindle constrained by an upper and lower set of rigid links attached to a vehicle chassis. Links typically called control-arms, a-arms, j-arms, or other names depending on specifics. We're primarily concerned with this style and its application to the front of off-road vehicles.

But first... A definition of terms are needed.

A link is a rigid physical body (or imaginary fixed distance) that connects two points, interchangeably called "bars". In kinematics we take the bars to be perfectly rigid; these do not stretch or buckle. A bar can be fixed in space, or it can be free. A point along a bar may be fixed in space, or it may be free. A bar that rotates about one of its points 360 degrees or more can be called a crank. A bar that rotates less than 360 degrees can be referred to as a rocker. Other connecting bars between cranks and rockers may be called couplers. Again, as far as kinematics are concerned, we don't care about the material properties of these components, only dimensionality.

One last, but important term to understand is DOF. Any linkage of bars and points may be characterized by the number of independent inputs to the system, called "degrees of freedom".

Why does any of this matter?

When viewing one side of a control-arm IFS design, the linkage becomes apparent. The upper control-arm, lower control-arm, spindle, and the frame between the arm pivots comprise a 4 bar linkage with one degree of freedom. The wheel hub moves the upper and lower control arms together, so we can consider the angle of the lower arm or the upper arm as the input for the system. Given the dimensions of the bars, we can solve for any angle in the system.

## Static Camber and the Camber-Travel Relationship

Camber is the angle a wheel makes relative to the ground. In most circumstances (relatively flat dirt road and paved track driving) the largest and best contact patch is formed when this angle formed is perpendicular. A vertical wheel on a flat surface produces the largest contact patch and the best grip. Consider a solid axle vehicle with wheels affixed to the axle such that the angle relative to the ground is 90 degrees. When driving in a straight line the contact patch is optimal, but quickly gets us into trouble while hard cornering. The carcass of the tire deflects under load, rolling the tire towards its edge. The weight transfer of the vehicle from inside to outside while cornering will deflect the the tires in such a way as to distort this contact patch for the worse, resulting in a loss of grip. To combat this effect, one could sacrifice straight line grip, by adding a bit of static camber to tilt the top of the wheels inwards (called negative camber). This design anticipates the corner, but this is sub-optimal. The tire might wear unevenly, and in a straight line the vehicle will have a less than perfect contact patch. On top of this, the degree to which camber is added, will only achieve an optimal contact patch for given level of cornering. Obviously one could out corner the amount of static camber, deflecting the carcass beyond this point. However, the idea works! The tire will deflect, but due to the static angle, a complimentary situation is achieved, resulting in a ~perpendicular wheel to ground angle while cornering. For this reason, a slight amount of static camber might used in applications like race cars that strive for extreme traction while cornering at high speed. Other than static camber adjustments, what if as the weight-transfer from the inside to the outside of the vehicle could produce a compression of the suspension such that the control-arm linkage increased the negative camber? This might effectively maintain a ~perpendicular angle of the tire while cornering. This elucidates an important consideration when designing static camber and the relationship between camber and suspension travel.

## The Tire's Edge and Arc Height

While picturing the x-y plane, place a bar at an angle of zero degrees along the horizontal. The horizontal distance from one end of the bar to the other is maximal, while the vertical distance between the two ends is minimal (zero). Rotating this bar about one end a few degrees from horizontal would produce a shorter displacement in terms of x, between the two points. This difference in displacement along the x-axis for a given angle change of the bar is referred to as the sagitta distance or arc-height. What does this mean for suspension linkages? If a vehicle with independent suspension is loaded (compression), the sagitta of the arcs created by the control arms will produce a lateral movement, causing the wheels to scrub outwards or inwards until settled. Likewise, if the suspension is unloaded (droop), the tires might scrub in the other direction. In either case, the cycling of the suspension causes the track width of the vehicle to change. To minimize this lateral scrubbing of the tires, we must minimize the sagitta. This can be accomplished by using as long a bar as possible in the linkage. The notion of sagitta is very important for solid axle vehicles which feature linkages to constrain their motion. Consider a panhard bar which locates a solid axle laterally beneath the vehicle. A compression or droop of such a suspension means the axle laterally moves out from underneath the vehicle as much as half the arc height. For larger amounts of suspension travel as seen in off-road vehicles, the resultant lateral movements of the tires is mitigated somewhat by loose soil and rolling tires.

Is there an optimal static position for suspension bars?

Yes, there is a position that could better than others. It can be beneficial to make the static angle of the suspension a specific setting. It depends on what one's goals are. This is done to control the direction in which lateral translation occurs from the initial static position. For a panhard linkage, this typically means statically mounting the pan hard bar ~parallel to the ground, or for an independent suspension design, keeping the control arms ~parallel to the ground. This is done so as to maintain the oscillations in the track width, or the lateral chassis translation around a particular position. In reality, desired ride height, spring rate, preload, or other factors could trump any concerns for these static angles.

Is there a relationship between travel, camber, and tire scrub?

The obvious answer is yes, but a better questions is, can one minimize the scrubbing of the tire that occurs due to lateral translation by changing the the camber of the wheel throughout the suspension travel? Some have designed the camber-travel relationship to be such that the contact patch will move to from center of the tire to an edge of the tire during full compression. This has the effect that for a straight compression of the independent suspension, the contact patches of the tires will shift from the centers to the edges opposite the direction of the arc height change to maintain contact with the same point on the ground, albeit for a significantly reduce contact patch. To some degree this could effectively "mitigate" tire scrub for compression of the suspension. In reality, these aspects of suspension design are not as critical in off-road settings and are often met with design constraints such as the size and dimensions of other vehicle components.

Given the camber-travel relationship, we know that the upper and lower bars of the independent suspension must be different lengths to achieve a change in camber through the travel. We also know that the upper bar must be shorter to tilt the top of the wheel inwards (negative camber) during compression. This characteristic is highly desirable for maintaining the best traction while cornering. If we assume that the pivots at the top and bottom of the spindle also form the steering pivots (as is the case on most control-arm independent suspensions), then projecting a line in the x-y plane through the pivots to the ground will be the steering axis (a 2D projection of it at least while we ignore caster angle). This angle projected into the ground through the spindle pivots has a technical name, and it is king pin inclination. This angle will intersect the ground at some point. The distance between this point and the center of the tire is called the scrub radius. If the king pin inclination intersects the ground inboard of the tire center line, the scrub radius is said to be positive, if it intersects out side the center-line of the tire it is said to be negative.

Why is this important?

There are several reasons. The first is that vehicle designers cannot always package the spindles, brakes, steering, and suspension link pivots within the wheel to achieve this. These components force the center-line of the wheel and tire to a more outboard position on the vehicle, resulting in a non-zero scrub radius. The second reason is that the more scrub radius the design has, the more feedback will be translated from the tire to the steering wheel. In off-road applications, this may be less desirable, but for on-road applications, this may be useful. Some drivers may prefer to "feel" the road surface. A third reason is simply be out of sacrifice. Many people prefer to reap the benefits of larger tires which can create a need for different wheel spacing, or even wheel spacers. Both of which will increase the scrub radius, bringing along all those things that come with it, but the benefits of wider tires may outweigh the need for optimal steering geometry. It is up to the designer to make such choices.

## Bump-steer and Roll-steer

Bump and roll-steer refer to the same phenomenon. That is, when a suspension system moves through its travel, the vehicle is steered. This is undesirable and can occur with independent and solid-axle suspension systems, and with the front or rear wheels. The reasons for this are the specific linkages of each and must be analyzed on an individual basis.

What causes control-arm IFS bump-steer?

This system's steering is typically driven by a rack-and-pinion, steering box and linkage, or other actuator.  From all of these, there must be some final link to drive the spindles to steer, usually termed "tie-rods".  To understand how steering can occur during suspension travel, consider that the control arms moving through the range of travel all have their own sagittas. The tie-rod link attached to the spindle will also have its own sagitta.  It should be obvious that if these sagittas differ in magnitude, the spindle will be forced to steer through the travel.

What?

Imagine that the upper control-arm is shorter than the lower. If we move the steering pivot  vertically (along the king pin axis) on the spindle, we get closer to either the upper or lower-control arm sagitta.  If one moves the steering pivot towards the upper control arm, and the upper control-arm sagitta is the same as the sagitta of the tie-rod, this will minimize the steering action on the spindle as the tie-rod and upper part of the spindle are closer to tracing the same sagitta. Likewise, if the lower control-arm has a longer sagitta than the tie-rod, moving the steering pivot on the spindle lower might increase the steering action during travel.

One may now start to realize that there is a point along the vertical (king pin axis) on the spindle between the upper and lower control-arm pivots, such that it traces the same sagitta as the tie-rod. This point would be the theoretical pivot location on the spindle for zero bump-steer.

Is that all?

No. In actuality, we may utilize more than just the position of the tie-rod pivot on the spindle to alter bump-steer characteristics. We can optimize the inboard steering pivot location as well as the overall tie-rod length, but for most situations modifying all of these parameters is costly.  Often, just changing the steering pivot location on the spindle and the tie-rod length is sufficient to minimize bump steer to a reasonable value.  To change the steering actuator location might require a complete redesign of a chassis rather than just the peripheral steering components.

It is all about making the relative differences in the sagittas of the links "play with each other".  This is the fundamental consideration for minimizing bump-steer in any suspension system.

## Solving the 4-Bar Linkage of the IFS Suspension

To solve the this linkage one must understand 2 concepts: vectors, and trigonometry ... and be proficient in algebra, none of which things can be briefly explained here, so we will summarize the process general below:

1. Vectors which start and end at the same point, form a "loop" and sum to zero.

2. Trace a path of vectors about the linkage starting from a point on one side of the linkage and ending at a point on the other side.  Notate the angles, signs, and magnitudes.

3. Write the equation for the vector loop.

4. Break the vector addition equation into its X and Y components equations.

5. List our knowns, unknowns, and what we will choose to be input to the system.

6. The linkage is now characterized by a system of two equations with two unknowns. If all dimensions are known, and all but two angles are known, we can solve for the remaining two angles.

7. We can now solve for the angles using the equations and much algebra, or we can plug them into a computer algebra system of choice.