Ackermann_steering_geometry

# Ackermann steering geometry **Ackermann steering geometry** is the arrangement of [[Steering|steering]] linkages that makes the inner and outer wheels of a steered axle turn through different angles, so that every wheel on the [[Vehicle|vehicle]] rolls about one common turn centre instead of being dragged sideways. It is named for [[Rudolph_Ackermann|Rudolph Ackermann]], who patented it in England on behalf of the Munich carriage builder [[Georg_Lankensperger|Georg Lankensperger]], and it is the reason a modern [[Car|car]]'s steering arms point inward rather than straight across. The geometry follows from one observation. In a turn, the inner front [[Wheel|wheel]] travels a smaller circle than the outer one, so it must be turned further; if both are turned by the same angle their axes cross at two different points and at least one [[Tire|tyre]] has to scrub. Placing the turn centre on the extended line of the rear axle, at a distance `R` from the centre of the car, the two front wheels of a vehicle with wheelbase `L` and track `t` must satisfy `tan δ_i = L/(R − t/2)` and `tan δ_o = L/(R + t/2)`. The linkage form of the same condition is simpler still: `cot δ_o − cot δ_i = t/L`, a constant, which is what a designer can actually build into a [[Four-bar_linkage|trapezium of fixed links]]. Pure Ackermann is a low-speed ideal. Once a vehicle is cornering fast enough for its tyres to develop slip angles, the wheels no longer point where they roll, and the steer angle the driver needs departs from the geometric one by an amount set by the tyres and the weight distribution — the understeer gradient. The framework microsim *Ackermann steering: two front wheels, one turn centre* draws the plan view and the handling chart side by side: the reader sets the turn radius, the wheelbase and the track, watches the two wheel axes and the rear axle meet at one point, and then shifts the weight split and the front-to-rear cornering stiffness ratio to see the steer-angle line tilt from understeer through neutral into oversteer. ## Advantages The first requirement of any steering geometry is that the tyres should not have to slip sideways to follow a curve.[^norris1906] A wheel rolls without scrub only when its axis passes through the centre of the circle it is travelling on; with rear wheels fixed, that centre must lie on the extension of the rear axle, and every other wheel's axis has to meet the same point. Turntable steering — a whole front axle pivoting on a central pin, as on a farm wagon — satisfies the condition trivially, because the axle itself swings to point at the centre. Its costs are the large fore-and-aft sweep of the wheels, heavy steering effort, and a bump under one wheel feeding straight back into the steering.[^norris1906] Ackermann's arrangement keeps the axle fixed and gives each wheel hub its own pivot close to the wheel, joined by steering arms to a tie rod, or track rod, that moves across the vehicle. With the wheels straight ahead the axle, the two arms and the tie rod form a trapezium; as the linkage moves, the inner wheel turns further than the outer one.[^norris1906] Because the pivots are close to the hubs there is very little fore-and-aft motion of the wheels, so the steering is lighter and far less disturbed by the road — the difference between a [[Rack_and_pinion|rack]] a driver can turn with one hand and a wagon's tiller. The numbers show how small the required difference is. On the microsim's default vehicle — a wheelbase of 2.7 m and a track of 1.5 m, turning about a centre 6 m from the middle of the rear axle — the inner wheel sits at `arctan(2.7/5.25)` = 27.2° and the outer at `arctan(2.7/6.75)` = 21.8°.[^sim-spec] The difference is 5.4°, and the single "bicycle" wheel of the simplified model would be at 24.2°, between the two. The linkage form checks on the same numbers: `cot 21.8° − cot 27.2°` = 2.500 − 1.944 = 0.556, which is `t/L` = 1.5/2.7. Steering more tightly widens the gap quickly — at a 3 m [[Turning_radius|turning radius]] the two angles are 50.2° and 35.8°, more than 14° apart — which is why the error in an approximate linkage matters most at full lock and hardly at all on the road. *Try: drag the turn radius from 50 m down to 5 m and watch the inner and outer readouts pull apart while the three wheel axes keep meeting at the single point O on the rear-axle line; then stretch the wheelbase from 2.7 m to 4 m and see both angles grow together as the geometry demands more lock for the same circle.*[^sim-spec] ## Design and choice of geometry A linkage of fixed-length links cannot satisfy the Ackermann condition at every steering angle, so practical design is a matter of choosing where to be exact and how large the error may be elsewhere. The classical approximation is to angle the steering arms inward so that their outer pivot points lie on lines drawn from each [[Kingpin_(automotive_part)|kingpin]] to the centre of the rear axle; the two arms are then joined by the tie rod, and the resulting trapezium tracks the ideal closely through the ordinary range of lock.[^norris1906] Moving the tie rod ahead of or behind the axle, shortening the arms, or changing their angle all shift the curve of actual difference against ideal difference, and a designer reads the result as a percentage: 100 percent Ackermann means the exact condition is met, 0 percent means the wheels stay parallel, and a negative percentage means the outer wheel turns further than the inner. The arrangement is older than the motor car, and its attribution is contested. It carries Ackermann's name from the English patent taken out on Lankensperger's behalf for horse-drawn carriages; [[Erasmus_Darwin|Erasmus Darwin]] had arrived at an equivalent arrangement for a carriage decades earlier, a prior claim set out in detail by Desmond King-Hele, who traces Darwin's drawings and the later patents together.[^kinghele2002] What the twentieth century added was not the geometry but the reasons to depart from it. ### Slip angles and the understeer gradient Modern cars are not built to pure Ackermann, because the geometry describes rolling wheels and a cornering tyre does not roll where it points. Each tyre develops a [[Slip_angle|slip angle]] proportional, at first, to the lateral force it carries, so the wheel that carries more load runs at a larger slip angle. The bicycle model condenses this into one line: the steer angle needed is the geometric angle plus a term proportional to lateral acceleration, `δ = L/R + K a_y`, where `K` is the understeer gradient — each axle's load divided by its cornering stiffness, front minus rear, `K = W_f/C_f − W_r/C_r`. A positive `K` is understeer and the driver adds lock with speed; zero is neutral; negative is oversteer.[^gillespie][^saej670] With the microsim's assumed inputs — a mass of 1500 kg, rear cornering stiffness of 60 kN/rad per axle and a front stiffness set as a ratio of it, all printed on the chart's sheet rather than hidden — a 55 percent front weight split at equal stiffnesses gives `K` = (0.55 − 0.45) × 1500 × 9.80665/60000 = 0.0245 rad/g, or 1.40° per g: mild understeer, and the driver adds about a degree and a half of lock per g of cornering.[^sim-spec] The same expression gives the characteristic speed, `v_char = √(L g/K)` = 32.9 m/s, or 118 km/h, the speed at which an understeering car needs twice its geometric steer angle. Move the split to 40 percent front and raise the front cornering stiffness by half, and `K` turns negative at −4.68° per g: the car now oversteers, the steer-angle line falls, and the critical speed `√(L g/−K)` = 18.0 m/s = 65 km/h is a real instability rather than a design target.[^sim-spec] *Try: in the understeer-and-oversteer variant hold the radius at 50 m and slide the front weight split down through 50 percent — the steer-angle line pivots from rising to falling, and when it crosses the zero line the readout hands you the critical speed, 65 km/h at a 40 percent split with the front tyres half again as stiff as the rear.*[^sim-spec] ### Reverse Ackermann and racing practice A racing car spends its important time at high lateral acceleration, where [[Weight_transfer|load transfer]] puts most of the front axle's work on the outer tyre, which therefore runs at the larger slip angle. Since it is the slip angles, not the geometric angles, that must be consistent with one turn centre, the wheel that needs the most steer is now the outer one, and some racing cars are built with reverse Ackermann for that reason: at speed it puts both front tyres closer to their best slip angle and keeps their temperatures down, at the price of scrubbing badly in slow manoeuvres.[^milliken1995] The choice is a statement about where the car earns its lap time. A formula car that never sees a car park can give up low-speed manoeuvring entirely; a rally car, which spends its life at full lock on loose surfaces, cannot. A road car, which parks every day and is judged on tyre wear, keeps positive Ackermann and accepts the small tyre scrub that remains at speed, where [[Cornering_force|cornering force]] rather than scrub decides the outcome anyway. ### Toe and the static setting Before any steering input, the two front wheels are set slightly non-parallel. Toe-in means the wheel planes converge ahead of the axle, toe-out that they diverge; the setting is small, quoted either as an angle per wheel or as the difference between the front and rear measurements across a pair of wheels. It interacts with Ackermann directly, because the trapezium's error at small angles and the static toe both decide which wheel is dragging when the car is nearly straight, and toe is the adjustment a workshop actually turns. The microsim's toe variant reads a 3 mm total toe-in across a 1.5 m track as `arctan(0.003/1.5)` = 0.11° per wheel. Two honesty notes come with that figure: the library's `toeAngle(toeIn, track)` divides the toe by the track, whereas toe is conventionally measured across the wheel rims, so the angle is smaller than a workshop's number for the same setting; and the angle on screen is drawn sixty times over, and labelled ILLUSTRATIVE on the sheet, because 0.11° is invisible at any honest scale.[^toe-spec] *Try: in the toe variant step the toe-in from 0 to 3 mm with the steering straight ahead — the wheel lines splay visibly because the drawing exaggerates them sixtyfold, while the readout stays at a tenth of a degree, which is the size of the real adjustment.*[^toe-spec] ## Extended Ackermann condition A vehicle towing a trailer has more axles to satisfy. The extended Ackermann condition requires that the [[Trailer_(vehicle)|trailer]]'s wheel axes, as well as the towing vehicle's, point at the same instantaneous turn centre; it is the criterion used when modelling the tow angle between an agricultural [[Tractor|tractor]] and its trailer.[^szakacs2010] In steady state the geometry is exact and easy to state: with the hitch over the tractor's rear axle, a trailer of wheelbase `L_t` whose axle points at the same centre runs on a circle of radius `√(R² − L_t²)`, where `R` is the radius of the tractor's rear axle. A tractor turning on a 15 m circle with a 12 m trailer behind it therefore puts the trailer's wheels on a 9 m circle — six metres inside the tractor's path. That difference is off-tracking, and it is the reason the condition matters outside the textbook. Everything a long combination hits, it hits with the trailer, on the inside of the turn; road geometry for a [[Semi-trailer_truck|semi-trailer truck]] is designed around the swept path rather than the tractor's path. During the transient, while the angle between the units is still changing, the condition is not met at all, and the trailer's wheels genuinely slip — which is what makes the tow-angle problem a question of [[Vehicle_dynamics|vehicle dynamics]] rather than a drawing exercise.[^szakacs2010] ## Minnesota *This section is specific to Wikitube.* Ackermann geometry sets how tightly a vehicle can turn, and a city's street design has to be drawn around the answer. The [[Minneapolis|Minneapolis]] Street Design Guide handles this with two vehicles rather than one. The design vehicle is "the least maneuverable vehicle expected to regularly use the intersection," and the geometry is drawn so that it can turn without crossing into opposing traffic; the control vehicle is "an infrequent but necessary user of the street," which may encroach on an adjacent lane or the gutter pan while it turns. Minneapolis names the DL-23 delivery vehicle, the SU-30 single-unit truck and the WB-40 as typical design vehicles, keeps an aerial fire truck as the control vehicle on most street types, and reserves the WB-62 semitrailer for truck routes and production and processing streets.[^mplsdesign] The arithmetic behind that distinction is the geometry above. A passenger car with a 2.7 m wheelbase at 35° of inner lock turns about a centre `2.7/tan 35°` = 3.9 m from its rear axle — a circle a residential corner absorbs without comment. A WB-62 cannot be steered to anything like that radius, and its trailer tracks inside whatever radius the tractor takes, so the corner either grows a large curb return, which lengthens every [[Pedestrian_crossing|pedestrian crossing]] on it, or the city accepts that the rare truck will use part of the opposing lane. [[Minnesota]] has both kinds of street in the same grid, which is why the guide separates the vehicle that sets the geometry from the vehicle that merely has to fit. The microsim is a plan-view model and should be read as one. Its wheel angles are exact geometry, but the handling chart beside them rests on assumed inputs — 1500 kg of mass, 60 kN/rad of rear cornering stiffness per axle and a front stiffness expressed as a ratio of it — which are printed on the sheet because they set the magnitude of the understeer gradient while the sliders set only its sign.[^sim-spec] Nothing in the model knows about the load transfer, the [[Camber_angle|camber]] change or the bushing compliance that decide a real vehicle's cornering stiffness; those belong to [[Car_suspension|the suspension]], and the number `K` is where the two subjects meet. ## See also - [[Understeer_and_oversteer]] - [[Automobile_handling]] - [[Toe_(automotive)]] - [[Car_suspension]] - [[Slip_angle]] - [[Steering]] - [[Differential_steering]] - [[Front_axle_assembly]] - [[Four-bar_linkage]] ## References [^norris1906]: Norris, William (1906). *Modern Steam Road Wagons*. London: Longmans, Green. Chapter "Steering," pp. 63–67 (the no-slip requirement, turntable steering and its drawbacks, the steering trapezium and the inner wheel turning further). https://archive.org/details/modernsteamroadw00norrrich [^kinghele2002]: King-Hele, Desmond (2002). "Erasmus Darwin's improved design for steering carriages—and cars." *Notes and Records of the Royal Society of London* 56 (1): 41–62. https://doi.org/10.1098/rsnr.2002.0166 [^gillespie]: Gillespie, Thomas D. (1992). *Fundamentals of Vehicle Dynamics*. Warrendale, Pennsylvania: Society of Automotive Engineers. Chapter 6, "Steady-State Cornering": the bicycle model, the understeer gradient `K = W_f/C_f − W_r/C_r`, and the characteristic and critical speeds. The Engineering portal's section plan cites this chapter as the standard form; page numbers are not pinned for this article. [^saej670]: SAE International. *Vehicle Dynamics Terminology*, SAE J670 (current revision J670_202206) — the standard definitions of understeer gradient, cornering stiffness and the associated sign conventions. https://www.sae.org/standards/j670_202206-vehicle-dynamics-terminology/ [^milliken1995]: Milliken, William F.; Milliken, Douglas L. (1995). *Race Car Vehicle Dynamics*. Warrendale, Pennsylvania: SAE International. ISBN 1-56091-526-9. Reverse Ackermann in the steering chapter (p. 715 as recalled; the page has not been re-checked for this article). [^szakacs2010]: Szakács, Tamás (2010). "Modelling and simulation of tow angle between agricultural tractors and trailers." *Landtechnik* 65 (3): 178–181. https://www.landtechnik-online.eu/landtechnik/article/view/2010-65-3-178-181/2010-65-3-178-181-en-pdf [^sim-spec]: Engineering portal pack, sim spec `specs/sims/Ackermann_steering_geometry.json` and its hooks: `design.vehicle.ackermann` (inner and outer angles and the bicycle-model angle, turn centre on the rear-axle line with `R` measured to the centre of the rear axle), `understeerGradient`, `steerAngle`, `characteristicSpeed`, `criticalSpeed` and `lateralAccel`. Build report, engineering run job E-G, September 18, 2026: at `R` = 6 m, `L` = 2.7 m, `t` = 1.5 m the readouts are 27.2° and 21.8°; at a 55 percent front split with equal cornering stiffnesses `K` = 0.02452 rad/g = 1.405°/g and the characteristic speed is 118 km/h; at a 40 percent split with `C_f/C_r` = 1.5 on a 50 m radius `K` = −4.68°/g and the critical speed is 65 km/h at 0.66 g. The report records nothing in the sim as ILLUSTRATIVE; the assumed inputs (1500 kg, `C_r` = 60 kN/rad per axle, `C_f` = ratio × `C_r`) are printed on the chart sheet and stated in the spec's sources. [^toe-spec]: Engineering portal pack, See-also variant `specs/variants/Toe_(automotive)` of the same sim. Its toe angle comes from `design.vehicle.toeAngle(toeIn, track) = arctan(toeIn/track)`, a convention the build report flags as unusual because toe is normally measured at the wheel rims rather than across the track; it is used as the library gives it and said so in the variant's sources. The drawn toe angle is exaggerated sixtyfold and marked ILLUSTRATIVE on the sheet; 3 mm across a 1.5 m track reads as 0.11° per wheel. Build report, engineering run job E-G, September 18, 2026. [^mplsdesign]: City of Minneapolis. *Minneapolis Street Design Guide*, §3.7B "Design and control vehicles": a design vehicle is "the least maneuverable vehicle expected to regularly use the intersection"; a control vehicle is "an infrequent but necessary user of the street" and may encroach into roughly one-third of an adjacent travel lane or into the gutter pan. Typical design vehicles are the DL-23, SU-30 and WB-40; the aerial fire truck and, on truck routes, the WB-62 are the control vehicles. https://sdg.minneapolismn.gov/design-guidance/intersections/design-and-control-vehicles  **Microsim — three.js (Wikitube framework):** *Ackermann steering geometry* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Ackermann_steering_geometry.html" data-title="Ackermann steering geometry"></div> *Built from `MICROSIM_GUIDE/specs/sims/Ackermann_steering_geometry.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Ackermann_steering_geometry) : [Wikitube](https://en.wikitube.io/wiki/Ackermann_steering_geometry) - skeleton pinned to revision 1371299735 (2026-09-18).