Rough/Soft-Field Space Landings

We have seen some tip-over failures among landers recently sent to the moon.  Looking at photos of these craft,  I see a pattern:  those that failed were too tall and narrow to reliably survive rough field landings in soft regolith on the moon.  There is a design criterion for that,  which worked well for Surveyor 1 and 3,  and the Apollo lunar landers,  1966-1972.  

There is much more to this than simple static stability,  per the classic high school physics problem!  There are the off-plumb angle effects of rough surfaces,  with both local slopes and obstructions.  There are the dynamic effects of nonzero vertical and horizontal speeds at the moment of touchdown.  There is also the issue of landing legs exerting too much pressure on the regolith and sinking-in.  That also leads to off-plumb angles,  as it will never occur symmetrically among the legs!   It also affects any subsequent takeoff,  as a sunken landing pad carries a load of regolith as extra weight,  and exerts a “tent stake” retardation.

The illustration shows all this empirical knowledge in one place.  Important are the minimum dimension s_min  across the polygon made by the centers of the landing pads,  and the height of the center of gravity h_cg.  The old criterion is shown,  which also helps address horizontal velocity limits at touchdown.  There is a lower limit on total pad area,  to avoid landing pads sinking into the regolith.  Tipping pads inward can help to “skitter-over” small rocks and protuberances.

The old criterion is more stringent than simple static stability on smooth level ground would indicate.  This is important because on the moon and Mars (and pretty much anywhere else with a solid surface),  the ground is neither smooth nor level.  There are plenty of obstructions,  and plenty of opportunities to dig a landing pad into the regolith,  and literally “trip” the craft into toppling over,  if it has any horizontal speed at touchdown.

For pressures exerted upon the regolith at touchdown,  there are empirical dynamic factors that get applied to the local weight,  which model the effects of small (but nonzero) vertical velocity a touchdown,  plus the possibility of one landing pad striking first.  These factors are only 1 at takeoff,  because the craft is not moving until it launches.

Overall Design Requirements

Details and Derivations

Basic static and dynamic stability is shown in the next illustration.  For the classic high school physics statics problem,  the weight vector “hanging” from the center of gravity must fall within the “footprint” defined by the landing pads,  or it will tip over.  Half the minimum footprint dimension,  and the center of gravity height,  lets one set up and solve a simple right triangle for the critical “out-of-plumb” angle that causes tip-over.  (For an odd number of legs,  go directly to the minimum half-dimension.)

If the effective local slope exceeds this critical angle,  the craft will topple!  Besides the general large-scale slope,  the small-scale roughness of the ground (or an obstruction under one landing pad) can make the local small-scale slope large indeed!  There is a table in the illustration of computed values from small critical angles at “tall-and-narrow”,  to very large critical angles at “short-and-squat”.  This large angle effect is exactly why the old design criterion of 1.5 < s_min/h_cg < 2 corresponds to “short-and-squat”.

Dynamic effects include both (1) horizontal speed at touchdown,  and (2) vertical speed at touchdown.  Nonzero horizontal speed brings with it the possibility of a landing pad on the leading side either striking a fixed obstruction,  or digging into the soft regolith and suddenly stopping.  Either way,  the retarding force is well below the center of gravity,  where the momentum vector is located,  thus creating an overturn torque.

Basically,  there is a radius from the center-of-gravity to the location of the blocked landing pad.  The craft rotates about the blockage point under the influence of the overturning torque,  which raises its center-of-gravity by the amount shown.  That radius gone vertical is the critical point:  any further and it topples over.  Thus,  conservation of energy says that if the horizontal kinetic energy exceeds the increase in potential energy,  it will topple over.

Static Stability and Dynamics Due to Horizontal Speed

Vertical speed at touchdown influences the transient pressure upon the regolith exerted by the pads.  If this transiently-higher-than-static pressure exceeds the failure bearing pressure of the regolith,  the pad sinks in!  Murphy’s Law says that will never,  ever happen evenly among the landing pads,  thus inherently acting to tip the vehicle out of plumb! 

Having a load of regolith atop a pad (or pads) is only an issue if the craft must later take off,  but bear in mind that Murphy’s Law also says those same regolith weights will be unevenly distributed among those pads.  That leads to attitude control problems,  in addition to the effects of simply being overweight at takeoff.

For low but non-zero impact speeds,  the usual empirical estimation practice in civil engineering is to double the static weight of the object,  as a measure of the transient force.  Add to that the strong likelihood of one pad striking first on rough ground,  and one should probably double that static weight again.  This is shown in the following illustration.

The average local static weight divided the total pad area is the static bearing pressure exerted upon the regolith below the pads.  “Factor 4 higher” (two factors of 2) would then be a good ballpark estimate of the transient pressure exerted underneath the pad that strikes first.  To avoid sinking in,  this transiently-exerted pressure must be less than the failure bearing pressure for that regolith. 

Problem:  we have little or no test data for the actual failure bearing pressure,  for any surfaces off of Earth. 

However,  we have observations from the moon and Mars that the regoliths there resemble the “soft fine dry sand” in Earthly sand dunes,  as long as any rocks embedded in that regolith do not touch each other.  Without touching,  those rocks simply cannot reinforce the strength of that regolith.  Essentially all of the surfaces of the moon,  and the vast majority of Mars,  meet that “soft fine dry sand” description,  as near as we know so far.

There are published strengths for various kinds of Earthly soils and surfaces,  used in foundation design.  These take the form of allowable pressures to support very long-term exposures,  so as to prevent soil settling under the foundations.  Those allowable pressures are usually about factor-2 lower than the actual failure pressure,  sometimes factor 2.5. 

These observations and surrogate data support the criteria and values given in the illustration.  These are ballpark-correct,  and should suffice well,  until better local information becomes available.  

Dynamic Effects Due to Vertical Speed

Observations and Recommendations:

The recent “short-and-squat” Firefly Aerospace lunar lander is upright and operating.  The two Intuitive Machines lunar landers were both “tall-and-narrow”,  and both ended up on their sides,  unable to function properly,  if at all.  See the pattern?  One has to wonder if the Intuitive Machines people actually knew about the old Apollo-era “short-and-squat” criterion.  That being from so long ago,  they may not have known.

Variants of SpaceX’s “Starship” vehicle are being seriously proposed as landers upon unprepared surfaces for both the moon and Mars.  That vehicle is inherently tall-and-narrow,  much like the “Falcon” booster cores they routinely fly back and recover.  However,  those “Falcon” cores have never landed upon anything but a flat, smooth,  reinforced-concrete landing pad,  or a flat,  smooth,  hard steel deck!  Those legs simply won’t work in rough,  soft dirt!  SpaceX has no experiences at all yet,  with rough-field and soft-field landings.  So far,  their projected “Starship” designs for the moon and Mars reflect that lack.

If these or any other companies want to land craft on planetary body surfaces off of Earth,  then they need to use the design criteria that I posted here.  This is old-time pencil-and-paper engineering stuff,  originally from the slide rule days.  You will not likely find this incorporated into most computer codes,  simply because we did not do it that way back then,  and the code writers may have been too young to know about any of this. 

I would be happy to consult about these issues.  Please contact me if you need help.  

Update 3-14-2025:  I found some dimensions for the Intuitive Machines "Athena" lander (over 4.5 m tall and only 1.57 m wide),  and also some information that the altimeter failed on both "Athena" and the first lander "Odysseus" (of similar dimensions).  The altimeter failure is why both machines touched down with very significant vertical and horizontal velocities!  How does it zero the speed at touchdown if it does not know its altitude?  It cannot!

For those dimensions and using a side view photo of an Athena mockup,  I was able to crudely approximate center of gravity height hcg as about 2.6 m,  and the minimum footprint "diameter" of a 6-leg hexagon as about 1.4 m.  Thus the min-s/hcg ratio is only about 0.5,  with a critical static tip-over angle of only about 14 degrees.  

I consider that angle just too low for a heavily cratered region on the moon.  The craft could touch down successfully at zero speeds,  and still tip over,  just because the local slope  near a crater rim was higher than 14 degrees!  Something quite common on the moon,  as we already know from Apollo.

Going further,  the radius from cg to the 2-pad leading contact point is about 2.7 m,  which at the critical 14 degree tip angle angle (radius gone vertical) says the cg height increase at tip-over is only about 0.09 m.  Using lunar gravity and that small height increase,  conservation of energy says the max survivable horizontal touchdown velocity is only about 0.5 m/s!  

Which more than likely is a limit that both machines failed,  and by a very large amount,  trying to land without an accurate altitude reading!  

But even with an altitude reading by which to nearly zero the vertical and horizontal speeds at touchdown,  the small critical static tip-over angle of 14 degrees may well be insufficient in lunar cratered terrain,  where local slopes can be much higher near crater rims.

"Short-and-squat" is just a more reliable lunar lander approach!  The numbers say so.  So does the prior experience decades ago.  You cannot argue with physics!  People are entitled to their opinions,  but no one is entitled to their own facts!