Dr. Dave's answers to frequently-asked questions (FAQs), mostly from the AZB discussion forum
for more information, see Section 4.04 in The Illustrated Principles of Pool and Billiards
and Vol. II of the Video Encyclopedia of Pool Shots
How can I easily adjust my aim to account for squirt (cue-ball deflection)?
"Squirt - Part IV: BHE, FHE, and pivot-length calibration" (BD, November, 2007) and "Throw - Part X: the big picture" (BD, May, 2007) cover aim-and-pivot techniques, which can be used to adjust your aim for squirt.
For more information, see the aim compensation for squirt, swerve, and throw resource page.
bridge length effects
Does the bridge length or tightness have any affect on squirt (cue-ball deflection)?
No, unless the bridge length is really short and the bridge fingers are very bony and have an extremely tight (i.e., non human) grip around the cue. This reasons for this are similar to the reasons why the grip can have no practical effect during tip contact.
Even if the bridge were perfectly rigid, it would still have absolutely no effect for bridge lengths beyond about 6-8 inches. The following video (at the 2:32 point point in part 2) shows and explains why visually: NV B.96 - Grip and bridge technique and advice.
And Diagram 4 in the following article gives some additional experimental proof related to endmass:
"Squirt - Part VII: cue test machine results" (BD, February, 2008)
Only mass and support within the 6-8" closest to the tip can make a difference, with the mass or support farther from the tip (within the 6-8") having less effect.
cue elevation effects
What effect does cue elevation have on squirt or cue ball deflection?
With more cue elevation, there is more swerve. Also, some of the swerve occurs immediately as the CB bounces off the table with the downward hit. I like to call this "immediate swerve." This effectively reduces the amount of "effective squirt."
Squirt really isn't that important alone (except for near-level-cue shots at fast speed or short distance, where swerve is not a significant factor). What is really important, especially with increasing cue elevation, is squerve (the combined effect of squirt and swerve, AKA net CB deflection). And this varies a lot with speed and conditions. And with higher cue elevations, the swerve effect dominates the squirt effect to the point where the squirt can be realistically ignored.
The vertical tip position also makes a difference. For more info, see tip contact height (follow/draw) effects.
endmass and stiffness
How does shaft endmass affect squirt (cue ball deflection) and how is endmass related to stiffness?
The characteristic that makes an LD shaft have less squirt (cue ball deflection) is reduced "endmass." See Diagram 4 in "Squirt - Part VII: cue test machine results" (BD, February, 2008). People who think extra stiffness is required to produce more squirt are incorrect. Added endmass alone (without added stiffness) produces significant increases in squirt. This supports the theory in TP A.31. The squirt of a shaft can be lowered by reducing the weight of the last 5-8 inches. This can be done by reducing the shaft's diameter, drilling out the core of the end of the shaft, using a lighter and/or harder tip (for more info, see cue tip hardness effects), and/or using a lighter (or no) ferrule. As demonstrated with the experiment in the article, mass closer to the tip has a greater effect on "effective endmass" than mass farther from the tip because it is moving more during tip contact (see what causes squirt), and beyond a certain distance, added mass has no effect at all. Here are cross-sections through common LD shafts (Predator 314 and the carbon fiber Revo) illustrating how the endmass is reduced:
Endmass is also related to lateral shaft stiffness. Firstly, a stiffer shaft will typically be thicker and heavier at the end, resulting in more weight at the end. Secondly, with a stiffer shaft, the transverse (or lateral or shear) elastic wave will travel faster and farther down the shaft (from the tip) during the brief contact time between the tip and ball. The farther the wave travels, the larger the effective "endmass" will be, because more mass is being involved during contact with the ball. This effect can be clear with carbon-fiber shafts, where you would expect the end of the shaft to be much lighter (which tends to reduce "endmass"); however, because the end of the shaft can also be very stiff (which tends to increase effective "endmass"), the amount of squirt can be comparable to a wood shaft that might be little heavier at the end. Another potential issue with carbon-fiber shafts is that they don't flex as much during and after a hit, so when you apply extreme spin (side, bottom or top), where the CB doesn't move away from the tip as quickly, there is a chance for a double-hit (which won't be directly noticeable, but the CB will appear to deflect or squirt more than expected). A wood shaft flexes more giving the CB room to clear away from the tip after the hit. If the end of the shaft is too stiff, this doesn't happen as well and a double hit can occur at large tip offsets. For related info, see:
"Coriolis was brilliant ... but he didn't have a high-speed camera - Part IV: maximum cue tip offset" (BD, October, 2005)
"squirt," "deflection," "stiffness"
cue vibration resource page
maximum sidespin resource page
Even though shaft stiffness has an indirect effect on squirt by affecting the effective endmass as described above, shaft stiffness does not directly affect cue ball deflection as many people think. In other words, the shaft is not flexing enough during tip contact to directly create enough sideways force to have a significant effect. Per the what causes squirt resource page, it is endmass (not shaft stiffness) that creates squirt. Some measurements and calculations showing that shaft stiffness has no significant direct effect on squirt can be found in the following analysis:
TP B.19 - Comparison of cue ball deflection (squirt) "endmass" and stiffness effects
Tip hardness can also have an effect on effective endmass because a harder tip will have a slightly shorter contact time. Because the transverse (or lateral or shear) elastic wave won't travel down the shaft as far during contact with a harder tip, the effective endmass and squirt can be less. However, a harder tip can also be denser and heavier, which would increase effective endmass and squirt.
For more information, see:
NV D.15 - Cue and Tip Testing for Cue Ball Deflection (Squirt)
NV B.32 - Squirt and the effects of endmass
NV B.1 - Mike Page's squirt and swerve video
"Return of the squirt robot" (BD, August, 2008)
HSV B.47 - effect of shaft endmass and squirt on miscue limit (for how the amount of squirt can affect the miscue limit)
what causes squirt?
tip hardness effects
tip contact time
Here's a list of advantages and disadvantages of low-squirt shafts.
What affect does shaft taper and the butt have on CB deflection?
If you make the tip end of a shaft (the last 5-8 inches) lighter, squirt is reduced. And if you change the mass or stiffness of the shaft beyond 5-8 inches, it has absolutely no effect on squirt.
Now, one must be careful to not confuse "squirt" with "the combined effects of squirt and swerve" (AKA "net CB deflection" or "squerve"). Swerve is affected by many factors including the speed of the stroke, the weight of the cue, the efficiency of the tip, the elevation of the cue, the properties of the ball and cloth, etc. Swerve (and not squirt) is what really makes aiming with sidespin over a wide range of shots challenging (and fun). So the taper and butt could affect "net CB deflection" or squerve.
Does a thicker shaft produce more squirt?
For a solid wood shaft of homogenous density, it is true that a thicker shaft will create more squirt, and for several reasons. The main reason is that the actual shaft mass close to the tip will be more. Also, the transverse stiffness will be larger. This creates two important effects. This will result in more sideways force as the shaft flexes (as the tip gets pushed sideways slightly as it rides the CB during the incredibly brief tip contact time). This is the effect that I show in my TP B.19 analysis to be extremely small. The other stiffness-related effect is that with a stiffer shaft, the speed of the elastic wave that travels down the shaft during tip contact will be faster, and this will increase the "effective endmass" of the shaft.
In your TP B.19 analysis, isn't a 1.6% (or 14%) difference significant to a top player, where small differences are important?
It is important to understand the implications of the 1.6% (or 14%) result. The 1.6% (or 14%) applies to the 1.8° of total squirt, so the amount of squirt due to the flex-force effect is a whopping 0.027 (or 0.25)° (1.5% [or 14%] of 1.8°)!!! This angular difference is extremely small and not even measurable in a practical sense. No human could possibly detect or create an angle change of that magnitude. And even if a cue produced significantly more squirt as the example in my analysis, the flex-force effect would probably still be too small to distinguish (not to mention that the cue would be much more difficult to use at a pool table).
In your TP B.19 analysis, how does the flex-force impulse relate to the sideways impulse between the tip and CB?
The forward impulse (F_imp) on the CB is:
F_imp = m_ball*v_fwd
where m_ball is the mass of the CB and v_fwd is the forward speed of the CB.
For a given squirt angle "α," the sideways impulse (S_imp) acting on the CB, which acts equal and opposite on the tip, is:
S_imp = F_imp*tan(α) = m_ball*v_side (1)
where v_side is the sideways speed of the CB.
The "effective endmass" feels this impulse, but it also feels an impulse resulting from a shear force resisting shaft bending because the flexed shaft is pushing back on the endmass (and an equal and opposite force is felt on the remainder of the cue). This is illustrated in the following diagram. The "effective endmass" of the shaft involves only the 5-8 inches of the shaft closest to the tip. It flexes and acquires momentum during impact. However, the transverse or lateral or shear elastic wave travels farther down the shaft during tip contact (10-16 inches) since the elastic wave must travel to and back from mass for it to be "felt" by the tip during contact. Therefore, a portion of the cue beyond the endmass is also flexed and also acquires some momentum during tip contact. For a slightly different explanation and illustration, see the quote from Jal below. Also notice in the diagram below how the tip end of the shaft also flexes back toward the CB. This is due to the off-center hit on the tip. As the tip grabs the CB, the contact forces bend the shaft end toward the CB as the CB pushes the tip and shaft away with CB rotation. This tip-end flex action is visible in the super-slow-motion videos on the cue vibration resource page, and the overall push of the endmass away from the CB is clear in the illustrations and video on the what causes squirt resource page.
Notice that the only way the CB can be deflected (squirted) down (in the diagram) is if the tip is pushing it down (in the diagram). If the tip is pushing the CB down (in the diagram), then the CB must also be pushing back on the tip up (in the diagram), as shown by the equal and opposite S_imp arrows. This is why the end of the shaft gains momentum (mv_end) that causes it to flex out. This flex out continues after tip release due to the momentum imparted during tip contact. Again, this action is clear in the super-slow-motion videos on the cue vibration resource page.
The momentum balance equation for the whole system is:
m_ball*v_side = m_end*v_end + m_beyond*v_beyond
where m_end is the effective endmass of the shaft, v_end is the effective speed developed by the endmass, m_beyond is the effective mass of the flexed portion of the shaft beyond the endmass, and v_beyond is the effective speed developed by this mass. Note that in a typical simplified analysis of squirt and endmass (e.g., in TP A.31), the flex effect is neglected, resulting in a slightly larger effective endmass. Here, because some of the CB sideways momentum (m_ball*v_side) is offset by the momentum beyond the endmass (m_beyond*v_beyond), the endmass will have slightly less momentum (and therefore a slightly less effective endmass than in the simplified analysis).
The proper momentum equation for the endmass, taking into account the shaft-flex effect, is:
S_imp - FL_imp = m_end*v_end (2)
where FL_imp is the impulse of the force associated with the flex of the shaft acting on the endmass (and equal and opposite on the portion of the shaft beyond the endmass) during tip contact with the CB.
Substituting S_imp from Equation 1 into Equation 2 gives:
m_ball*v_side = m_end*v_end + FL_imp
Therefore, it is clear that the CB's sideways momentum comes from two effects: the momentum transfer from the endmass (m_end*v_end) and the impulse of the flex force (FL_imp). Effective endmass is affected by lateral or transverse stiffness (as described in the main section above), but force due to flexing is a separate effect. This is clear because you can have one without the other. If most of the endmass were in the tip and ferrule, and the shaft had little or no stiffness (i.e., if it took little or no force to flex the shaft), there would still be the "m_end*v_end" effect but little or no FL_imp effect. And if the shaft end were stiff laterally but had negligible endmass (even though a greater length of the shaft would contribute to endmass), there would still be a "FL_imp" effect but little or no "m_end*v_end" effect. When the CB pushes sideways on the tip, it creates endmass momentum, but it also flexes the cue. Both of these things require force, hence the two terms in the equation above.
In the TP B.19 analysis, I am comparing the peak force involved with S_imp to the peak force involved with FL_imp. The flex effect is shown to be very small in comparison to the endmass effect.
There is a little "smoke and mirrors" going on here due to the awkward definition of "effective endmass" and because the flex force is actually a dynamic and distributed force acting along the "endmass" and beyond. Also, my static measurement of flex force and deflection doesn't perfectly model the shaft flex involved with the sideways endmass motion. The effective length of the flex during tip contact is shorter than in a static deflection test (so the flex force will be larger). Although, with the Predator Z-2, the tip end of the shaft is very flexible with its small diameter (11.75mm) and hollow core (about 5 inches long), so a lot of the flex in the static case also occurs closer to the tip than the joint. Regardless, the analysis does provide good ball-park numbers.
Why does the end of the shaft flex close to the tip as shown in the diagram above?
The diagram below illustrates this. The force between the tip and CB acts equal and opposite on each object. The CB speed is in the direction of this total force. The CB spin is created since the force's line of action acts at a tip offset relative to CB center, creating a moment or torque. The equal and opposite force on the cue tip also acts off-center on the end of the shaft, creating a bending moment or torque. This is what causes the end of the shaft to flex.
Notice how the force from the tip pushes in the direction the CB heads (i.e., the force arrow is parallel to the speed arrow). This force has two components. One component acts to the right (in the diagram) on the CB and to the left (in the diagram) on the tip. This component is what pushes the CB forward in the line of aim of the shot, and causes the end of the shaft to flex as shown in the diagram (because the force acts off center on the tip). The other force component acts down (in the diagram) on the CB and up (in the diagram) on the tip. This component is what causes CB deflection (squirt) and causes the end of the shaft to gain momentum (speed) away from the CB (up in the diagram) that causes the shaft to flex out. This outward flex starts during tip contact (as the CB rotates during contact and pushes the tip away, as described and illustrated on the what causes squirt resource page) and continues after tip release due to the momentum imparted during tip contact.
How does the force between the tip and CB change during tip contact?
A tip behaves like a spring, where force is directly related to compression (F = kx). If the tip is compressed a little, it is because there is a small force acting on it. If it is compressed a lot, it is because there is a large force acting on it. When the tip first touches the CB, there is no compression at all, and no force. As the cue continues to move forward, the tip begins to compress gradually. This results in an equal and opposite force on the tip and CB that also increases gradually. The CB begins to accelerate as soon as there is a force acting on it (F = ma). The CB accelerates slowly at first since the force is small, but as the tip compresses more and more, the CB accelerates faster and faster. When the tip reaches full compression, the force between the tip and CB is at its peak, and the CB is accelerating at its fastest rate. As the CB begins to move faster than the tip, the tip begins to spring back and the force between the tip and CB starts to decrease in proportion to how much it is still compressed. The force is maximum at the beginning of the decompression and is zero after the tip has fully decompressed (as it is about to release from the CB), but there is force acting between the tip and CB during the entire spring-back or decompression stage. The force continues to cause the CB to accelerate forward this entire time (until the tip totally releases from the CB).
How can the tip and shaft end move away from the CB laterally if the CB is already gone when the tip and shaft move?
Acceleration (an increase in speed) can occur only with force; therefore, the CB can accelerate only while force is acting from the tip. Likewise, the tip and shaft can't be given lateral speed, unless lateral forces are acting (during tip contact). The tip/shaft moves away laterally after the tip leaves the CB because the tip/shaft was given lateral speed during contact (while lateral forces were acting). It is the momentum (mass * speed) of the endmass that makes the tip and shaft move away from the CB after impact. Now, any flex energy still stored in the end of the shaft after tip release will also cause some post-release vibration (as is evident in the slow motion videos on the cue vibration resource page), but this is independent of (and much smaller in amplitude than) the larger-scale tip/shaft motion away from the ball (due to the endmass lateral momentum), as is evident on the what causes squirt and cue vibration pages. Below are some stills from a super-slow-motion video showing of motion of the tip, CB, and shaft evolve during and just after contact. The red line marks the initial position of the top of the shaft and initial tip contact point, and the red dot marks the initial position of a distinct point on the ball. They are in the same positions in each successive image. It is clear that the tip and shaft move down (laterally) away from the CB as the CB moves forward with increasing speed and spin during the entire period of tip contact.
from Jal (in AZB post):
As the transverse (shear) wave travels down the shaft, it has to be reflected back off the medium in which it's propagating and reach the tip again in order to be felt by the cueball. If you divide the shaft into thin sections, when the wave reaches an interface between two sections, the leading section first transmits the shear force to the trailing section. The trailing section then, naturally (Newton's Third), produces a reaction force, equal an opposite, which then travels as a wave (disturbance) back toward the tip. Here's sort of an illustration with the black arrows indicating the original shear and the red ones indicating the reactions.
Note: Maybe I should have reversed the locations of the black and red arrows in each section for better clarity.
As you get farther down the shaft, there isn't enough time for the reflected waves to return all the way to the tip. This is indicated by the pale red arrows. However, in this region, the sections are 1) moving to the left (have momentum), and 2) the reaction forces they're generating, while not reaching the tip, are slowing down the upper section whose reaction forces are reaching the tip.
Therefore, since we define "endmass" to be the mass that, from being pushed aside, produces a counter-force that acts on the cueball, only the upper section fits the bill. It (the upper section), being slowed by the reaction forces from the lower section, does not posses all the equal-and-opposite momentum of the cueball - the lower section carries part of it. Thus, there must be more force acting on the cueball than we can account for from the endmass (proper) alone. That force must be the restorative (flexing) force generated by the bend, since the lower section can't contribute - its waves never reach the tip.
How can you be so sure shaft stiffness and flex has a direct effect on squirt?
1.) Shaft endmass can be increased significantly by adding mass close to the tip (e.g., by using a heavier tip or ferrule, by inserting something heavy but not stiff into a cored-out shaft, or by physically adding external mass to the tip end of the shaft). This has been clearly demonstrated with numerous experiments by me, Mike Page, and others. In these cases, the endmass is increased dramatically with no increase in shaft stiffness.
2.) Removing endmass from a shaft without significantly changing the stiffness of the shaft (e.g., by using a lighter ferrule and/or by drilling out the core of the shaft end), can significantly reduce the amount of effective endmass and squirt. This has also been demonstrated with the design of Predator shafts. Drilling out the core does not have a significant effect on shaft-end stiffness, but it does dramatically reduce endmass and squirt (as does the lighter ferrule).
3.) As my TP B.19 analysis shows, the total sideways force acting between the tip and CB is due to two physical effects. Part of the force contributes impulse which imparts momentum to the endmass of the shaft. The other part of the force (much smaller) is required to flex the shaft. The end of the shaft effectively looks like a mass supported by a spring. Think of a simple diagram of a linear spring-mass system with an applied force pushing on a mass supported by a spring. Some of the force applied to the mass goes into accelerating the mass (imparting momentum), and some goes into compressing the spring (i.e., F_applied = ma + kx). The resultant force experienced by the mass is not the force applied to the mass (F_applied); rather, it is the amount of excess force not being resisted by the spring (F_applied - kx). As described above, this same logic applies to the squirt-endmass-stiffness problem. The difference between the shaft-endmass-lateral-stiffness problem and the simple linear-spring-mass problem is that the total endmass of the shaft depends on shaft stiffness (in addition to the weight of the tip, ferrule, and anything else on or in the end of the shaft), but the spring force is still there.
How does one go about measuring shaft transverse stiffness (e.g., to compare two different shafts)?
To measure the "transverse stiffness" important in discussions concerning squirt (CB deflection) and endmass, rigidly support the entire cue on a table (with clamps and/or heavy weights) so only the portion "active" during tip contact is hanging over the edge of the table. "Endmass" involves only the 5-8 inches of the shaft closest to the tip, but the transverse wave travels farther down the shaft during tip contact (10-16 inches) since the transverse wave must travel to and back from mass for it to be "felt" by the tip during contact, so I suggest 8 inches (as a good average value of flex length during tip contact). Then hang a weight from the tip and measure how much the tip moves down (i.e., how much the shaft end flexes). The transverse stiffness is:
k_trans = (weight applied to tip) / (distance tip moves down)
See TP B.19 for an example calculation (using the measurement I took with a Predator Z2 shaft).
The shaft transverse stiffness is inversely proportional to the amount the shaft flexes. With a smaller stiffness, the shaft flexes more; and with a larger stiffness, the shaft flexes less. It is important to not confuse the "deflection" of the shaft (how much the tip moves down when the weight is applied) with the CB "deflection" caused by the shaft. A shaft that "deflects" more will usually produce less CB "deflection" (squirt), unless the tip and/or ferrule and/or shaft end are heavy.
The "stiffness" or "whippiness" a player "feels" applies more to the entire cue. That could be quantified similarly by extending more of the cue over the edge of the table or by measuring frequencies (rates) of vibration of the cue during a hit (e.g., with an accelerometer). Although, there are many factors that affect the "hit" or "feel" of a cue.
The Meucci shaft over the years has had features to reduce the endmass:
1. The ferrule has always been thin walled relative to most other cues. (the plastics used in ferrules is usually of higher density than maple)
2. The ferrule has been made of a less dense material than most other ferrules on competing cues.
3. On recent shafts (black dot), the tenon has been tapered like the end of a pencil (not that extreme), yet the internal walls of the ferrule have remained cylindrical. This further reduces endmass by introducing a tapered hollow region right behind the tip.
Here's a photo from Cue Crazy (in AZB post) relating to "isuedtoberich's" quote above, called Meucci's Power Piston design:
How much of an effect does added or removed endmass have on the resulting squirt of a shaft?
Based on the theory in TP A.31 and the data in "Squirt - Part VII: cue test machine results" (BD, February, 2008), a typical cue might have a ball-to-endmass ratio of about 30, corresponding to an effective endmass of about 5 grams. Any endmass added to or taken away from this would affect the amount of squirt proportionally. For example, for the 0.3 gram and 1.3 gram added massés in Diagram 4 of the article, the total endmass would be 5.3 with 0.3 grams added close to the tip and would be 6.3 with 1.3g added close to the tip. This comparison corresponds to an endmass ratio of 6.3/5.3=1.2. The robot measurements for squirt angle were 3.9° with the larger added mass and 3.3° with the smaller added mass. This is directly related to the endmass ratio: 3.9/3.3 = 6.3/5.3 = 1.2.
from Bob Jewett (in AZB post):
The stick transfers energy to the cue ball by compressing like a spring along its whole length. The compression wave happens at the speed of sound in the stick, which is about 13000 feet per second. This speed is the fastest that the butt can learn of something colliding with the tip. Some people make the mistake of thinking of the cue stick as being perfectly rigid and incompressible, but it's not. So, the shot proceeds like this: the stick is coming forward and the tip meets the ball. The tip starts to compress, force and acceleration of the cue ball start to build up. The ball also starts to compress, since it too is not incompressible. The ball has started to move, but is not up to the speed of the stick yet, and the stick has started to slow down as its energy is transferred to the cue ball. This continues until the tip (and ferrule and joint and butt) reach maximum compression along the length. At this exact point some amazing things are happening. The stick and ball are moving at the same speed. The force between stick and ball are at their maximum. The compression along the length of the stick (including the tip) is at its maximum. The energy stored in the spring-like compression of the tip (and stick and ball) are at their maximum. For a typical ball and stick, the speeds of the ball and stick are 75% of the original stick speed.
After this point of maximum compression, the ball is pushed forward from the tip by the compression of system. The ball starts to move even faster from this force and the stick continues to slow down. This "unwinding" process continues until the ball finally leaves the tip. At that point, the ball is going at about 130% of the original stick speed, and the stick has slowed down to about 50% of its original speed. (The 130% would be 150%, but the tip is not perfect in springing back to its original shape, and energy is lost.)
Now the hand comes in. Human flesh makes a much "softer" spring than the leather of a tip or the wood that is compressed along the length of the stick. Think of the tip as about the stiffest car spring you can imagine and your hand like a rubber band. The cue ball is gone by the time your hand -- which is still moving forward at full speed -- can wind up even a little. As the hand winds up on the stick and relaxes, which takes about 20 milliseconds, the hand is slowed to about 80% of its initial speed and the stick goes from 50% back up to 80% of its initial speed. Of course this re-acceleration of the stick by your hand is useless in that the cue ball is long gone.
How does a heavier stick affect things? It changes that 130% number. The formula is in Byrne's Advanced book, and somewhere in my columns in Billiards Digest and certainly in Ron Shepard's paper and Dr. Dave's book. A heavier stick through the spring action, puts slightly more energy into the cue ball.
As for how the weight of the stick affects the squirt, I think the answer is that it doesn't, much. Squirt is caused by the spinning cue ball pushing the stick to the side during the contact time of an off-center hit. The amount of squirt is determined by the mass that is being pushed to the side. Since the stick is very floppy side-to-side (as compared to length-wise compression), only the front part of the stick can participate in the squirt during the 1 millisecond or so of contact time. A heavier stick will increase the contact time a little, and that will increase the squirt a little, but I think this effect is pretty small.
Phrased technically, the transverse wave has a very slow propagation velocity along the length of the stick, and so the joint and butt cannot participate in the sideways push that causes squirt.
You should find Mike Page's discussion of his experiment with vise grips on the shaft which determined how much of the shaft participates in squirt.
As Fred mentioned, a major problem with some of the Jacksonville Project was that Iron Willie had too stiff a grip -- like vise grips -- and too hard a bridge. I have heard that Predator's current cue testing robot has fixed those problems to hold the cue more like a human at both ends.
As for some of your other questions, in theory the squirt should depend on stiffness of the cue since that should change the speed of the transverse wave. In practice, "end mass" seems to be a much better indicator of squirt than stiffness. There are stiff cues with little squirt and stiff cues with lots of squirt. A major red herring along the path of squirt studies was the fact that carom cues tend to be stiff but have relatively low squirt. They usually have smaller tips than pool cues.
reply from Dr. Dave:
For those who want more information related to Bob's post, further descriptions, illustrations, and demonstrations can be found on the following resource pages:
what causes squirt?
endmass and stiffness effects
shaft squirt (CB deflection) testing
cue tip hardness effects
effects of light vs. tight grip
Concerning how the cue and CB speeds vary with tip offset from center, cue weight, and cue speed, that is covered in detail (with physics and math) in: TP A.30 - The effects of cue tip offset, cue weight, and cue speed on cue ball speed and spin.
low squirt (low deflection or LD) shafts
Can a low-deflection (LD) shaft help my game?
See low-squirt (low-deflection, LD) shafts.
Does the miscue limit depend on the shaft's squirt?
See: HSV B.47 - effect of shaft endmass and squirt on miscue limit. It appears that a cue with more endmass (a lot more in the video) allows greater tip offset. With more tip offset, you would expect to get more english. You would also expect to get more squirt than you would get even with the same endmass. If you watch all of the shots in the video, you will see that the cue with the added endmass had much more squirt than the cue without the added endmass, much more than can be explained by a small difference in tip offset. Also, with more squirt comes less english (for a given tip offset), because the effective offset is less. If you look at the stripe on the ball in the super-slow-motion clips, you will see that the CB actually has slightly more english (spin per distance) with the low-squirt cue (due to a larger "effective tip offset"), even though the "actual tip offset" is slightly greater with the added-endmass cue!
For more info, see:
Can the type or brand of chalk affect the amount of squirt?
All commercially available pool chalk, assuming the tip is holding it, grabs the CB without any slipping whatsoever. When the tip slips, a miscue results. Now, "partial" miscues are possible, where the tip mostly grabs and just slips a little. With any miscue (partial or full), there is significantly more squirt because the tip moves sideways more as it slides over the edge of the CB (see example videos here). With more tip sideways motion (which requires force), the CB will experience more equal-and-opposite-reaction sideways force, resulting in more CB squirt. Also, I would expect the amount of squirt would be very inconsistent if there were partial or full slip due to the complicated nature of impact-induced slip. That's why the tip probably doesn't slip with most shots, because with most shot (assuming the tip is well chalked), CB squirt is very consistent.
For more info concerning how different chalks compare, see chalk brand comparison results.
Where can I find published data on squirt (CB deflection) values for various cues?
Platinum Billiards did some tests a while back and posted a collection of extensive data (see below). Meucci has also done some testing measuring the combined effects of squirt, swerve, and throw, so no reliable squirt data is available (videos and results are available here). Ron Shepard's squirt paper reports a squirt angle range of about .5 to 2.3° for low- to high-squirt cues, corresponding to a pivot point range of about 50" to 10". Platinum's data (see below) ranges over 1.3 to 2.3° of squirt angle and 7.6" to 14.1" for pivot points. Some other data is available on the cue natural pivot length resource page, where the numbers seem to fall in between the ranges reported by Shepard and Platinum.
If you want to take your own squirt measurements, and you don't have access to a robotic cue-testing machine, the following videos offer advice and procedures for doing your own experiments:
For more info, see: "Cue Tip Squirt Testing" (BD, June, 2014) and "The Carbon Fiber Shaft Craze" (BD, December, 2019).
from Platinum Billiards (results from tests on a cue-testing robot called "Iron Willie"), circa about 2008:
HOW AND WHAT WE TEST
We ask the question "which shaft deflects least?" because the butt of the cue has little effect on cue ball deflection. However, shafts are generally tested on the same brand of butt and the test weight for all is kept close to 19 ounces. All shafts are tested as sold by the manufacturer including tip type and tip curvature as noted. All tests are performed using a robot which makes precisely the same stroke with each cue, and for this test the machine is set to produce cue ball speeds of around 15mph. A series of four shots is made with each cue and the resulting cue ball deflection is recorded on a target 50" away which is exactly the distance between the foot string and the head spot on a 4 ½ x 9 pool table. The four shots are 6mm (about ¼") and 12mm left of center, and 6mm and 12mm right of center, and these offsets are measured from the center of the cue ball to the center of the shaft. The actual cue ball deflection produced by each shot is measured and the average for the series is given in the chart below in millimeters and inches.
|Universal SmartShaft (Low Squirt)||dime||37.9||1.49||-8.6%||10.7||low|
|McDermott i-2||dime||38.6||1.52||-6.9%||10.5||med low|
|Universal SmartShaft (Regular Squirt)||dime||39.4||1.55||-5.0%||10.3||med low|
|McDermott i-1||dime||39.6||1.56||-4.4%||10.3||med low|
|Meucci Red Dot||dime||40.1||1.58||-3.2%||10.1||med low|
|Mezz Power Break 2||quarter||41.7||1.64||0.5%||9.7||medium|
|Cuetec SST||nickel||44.2||1.74||6.6%||9.0||med high|
|X Breaker||44.3||1.74||6.8%||9.0||med high|
|Meucci Black Dot||dime||44.4||1.75||7.2%||9.0||med high|
|Axiom J/B||dime||46.0||1.81||10.9%||8.7||med high|
|Bunjee Blaster||nickel||46.0||1.81||10.9%||8.7||med high|
|Lightning Bolt||46.2||1.82||11.4%||8.6||med high|
Platinum Billiards is an independent company and has no affiliation with any billiard product manufacturer. The performance information we provide is based on careful scientific testing and observation. We are highly experienced at testing the performance of cues and we believe that our methods are sound and accurate. However, we do not claim that our findings are absolute. We are aware that cues of a same model vary slightly and as we test more samples of each, the numbers will become more refined. If any manufacturer is unhappy with our results and/or feels that the ratings are unfair, we encourage them to contact us and we will be happy to answer questions about our methodology and/or arrange for the testing of any cues they would like to send us, and if warranted, we will adjust the numbers accordingly. We can only offer testing of cues, shafts, products that are currently on the market. We do not offer testing for prototypes or products that have yet to be made available to the general public.
If you are curious about a shaft or cue not on the list above, the easiest way to compare CB deflection is to determine the cue natural pivot length with simple tests. This can then be compared to the "pivot point" values reported above.
robot test results
Where can I find information on experimental results from squirt-testing robots?
See published data for some cue-comparison results from Platinum Billiards resulting from cue tests with "Iron Willie" (a cue-testing machine). The Jacksonville Project also did some work with "Iron Willie." Meucci has also done some tests with their "Myth Destroyer" machine (although, due to the violations of the "Rules of CB Deflection (Squirt) Testing" below, Meucci's setup should probably be called the "Myth Creator" instead).
Alexander Sorokin has also developed a cue-testing machine. More info can be found here: Cue Testing Unit.
The following articles document work with a cue-testing machine developed at Colorado State University (for CueStix International):
"Squirt - Part VII: cue test machine results" (BD, February, 2008)
"Return of the squirt robot" (BD, August, 2008)
The CSU machine used a spring stretched to various indexed and latched lengths to create repeatable cue speeds. The cue was supported by an adjustable-height V-bridge and a rubber-lined grip which slid on linear bearings. When a cue was loaded, both the grip and bridge heights were adjusted to ensure the tip was at center-ball level with the cue horizontal. The squirt angle was measured both manually by observing where the CB struck at the end of the test table, and electronically with two crossing IR beams that detected the CB speed components in two directions. Before going with the spring-loaded linear system, we had considered the three prototype design concepts demonstrated in the following videos:
Pneumatic cue-stick tester prototype
Spring-loaded cue-stick tester prototype
Motorized cue-stick tester prototype
The final machine built and used for testing was of much better quality than the prototypes. Results from lots of testing done with the final machine are available in the BD instructionial article links above.
NOTE - when using a machine to test cues, the "grip" needs to be flexible, like the flesh in a human hand (e.g., by lining the mechanical "grip: with silicone rubber).
The problem with a non-human, extremely-firm robot grip is that it can add significant effective weight to the cue. If the grip is totally rigid, the weight of the machine's "hand" and "arm" completely add to the weight of the cue. For example, if you put an 18 oz cue in a rigid machine grip, and the weight of the machine's "grip" is 20 oz, the cue will act like a 38 oz cue! The result of this is that the CB will not leave fast enough to clear the tip with an off-center hit. The tip will either remain in contact with the CB or catch up after initial contact, creating either a push or double hit. The hit will look and sound normal, but the CB will have more squirt (CB deflection) ... sometimes a lot more (as if there where a miscue). Lot's of care must be taken when using a machine to test and characterize cues that will be used by non-machine humans.
If you don't have access to a robotic squirt-testing machine, decent results can be obtained with careful experiments with human shooters. The following videos recommend procedures for how to do this:
For more info, see: "Cue Tip Squirt Testing" (BD, June, 2014) and "The Carbon Fiber Shaft Craze" (BD, December, 2019).
Rules of CB Deflection (Squirt) Testing
Things one must be aware of when testing a shaft or tip for cue ball deflection (squirt), using either a robot or a person, include the following:
from Cornerman (in AZB post):
Meucci's system starts with a false condition: too tight a grip, a grip that human couldn't possibly have, so when it strokes the cueball, the cuestick tip and cueball are in longer contact than the 0.001s measured contact time of a human stroke.
Contact time x speed of transverse wave propagation down the shaft = length of shaft "in effect" of the collision.
A normal human stroke will have something in the order of 6" of the front end of the shaft "in effect;" the Myth Destroyer, due to its overly tight grip, has a much longer length of shaft "in effect."
More shaft length in effect = more mass in effect = more squirt.
Meucci's ferrule design [see below] was meant to partially delay or de-couple the ferrule from the shaft (there's some space between the top of the ferrule to the top of the tenon), so it does well for the Myth Destroyer. Predator's design is significantly reducing the overall mass in the first 6", and it wasn't meant to beat a non-real robot situation. So Predator does worse on the Myth Destroyer (because the Myth Destroyer wants to add mass that's beyond the 6" if the ferrule isn't decoupled).
And that's just the robot grip problem.
Does squirt change with speed?
"Cue ball deflection" or "squirt" refers to the angular deflection of the CB immediately off the tip. Squirt does not vary with speed. Proof, from careful experiments with cue-testing robots, can be found here:
"Squirt - Part VII: cue test machine results" (Billiards Digest, February, 2008)
"Squirt - Part II: experimental results" (Billiards Digest, September, 2007)Now, for most shots at a pool table (where the cue must be elevated some to clear the rails), with english comes both squirt and swerve (CB curving). And swerve does vary with speed (and with conditions and cue elevation). So the combined effects of squirt and swerve (AKA "squerve" or "effective deflection" or "effective squirt") does vary with speed. With a slow shot, the swerve happens quickly over a short distance, and this reduces the squerve of the shot. With a faster shot, the swerve is delayed and the squerve is larger. Here's a good demo of this effect:
NV A.17 - Effective squirt vs. speed
And here's another from Vol. II of the Video Encyclopedia of Pool Shots demonstrating the combined effects of squirt and swerve:
Again, squirt doesn't vary with speed, but swerve and squerve do.
squirt, swerve, and throw confusion
What is squirt (CB deflection) and how is it different from swerve (CB curve)?
From the online glossary:
squirt (AKA "cue ball deflection"): angular displacement of the cue ball’s path away from the cue stroking direction caused by the use of sidespin.
swerve: curve of the cue ball’s path while sliding due to cue elevation and sidespin.
cue ball deflection: same as "squirt;" also sometimes used to describe the net effect of squirt and swerve (AKA "squerve" or "effective squirt" or "net cue ball deflection").
net cue ball deflection (AKA "squerve" or "effective squirt"): the net effect of "squirt" and "swerve" (i.e., the cue ball deflection off the aiming line at object ball impact).
effective squirt (AKA "squerve"): same as "net cue ball deflection."
Total or net or effective CB deflection is the end result of of both squirt (sometimes also called "deflection") and swerve. Throw also affects some shots (some more than others). A complete summary and demonstration of all squirt, swerve, and throw effects can be found here:
complete summary of all squirt, swerve, and throw effects (with supporting resources)
Squirt depends on the amount of sidespin used and the properties (endmass) of the shaft. Swerve depends on shot speed, shot distance, the amount of sidespin, cue elevation, and ball/cloth conditions. See the effects summary resource page for explanations and demonstrations. Vertical tip position (for draw and follow) also affect squirt and swerve. For more info, see: squirt and swerve draw and follow effects. Again, net CB deflection is a result of the combined effects of squirt and swerve.
Here are some video demonstrations and explanations of squirt, swerve, and throw:
How can you predict the directions and amounts of squirt, swerve, and throw with various types of shots?
Here's a diagram from "Squirt - Part I: introduction" (BD, August, 2007) that shows all of the effects involved with using sidespin:
Some people might think the throw direction in the diagram is wrong due to collision- or cut-induced throw (CIT). Think about it yourself and decide if you think the diagram is correct or not. Many people seem to be confused by the separate effects of squirt and swerve. Diagram 4 from the article (see below) helps clarify things.
The phrase "effective squirt" or "net cue ball deflection" is used for the combined effects of both squirt and swerve. The term "squerve" (SQUirt + swERVE) means the same thing. The following series of instructional articles dealing with squirt covers all of the details of squirt and swerve:
"Squirt - Part I: introduction" (BD, August, 2007).
"Squirt - Part II: experimental results" (BD, September, 2007).
"Squirt - Part III: follow/draw squirt and swerve" (BD, October, 2007).
"Squirt - Part IV: BHE, FHE, and pivot-length calibration" (BD, November, 2007).
"Squirt - Part V: low-squirt cues" (BD, December, 2007).
"Squirt - Part VI: tip shape" (BD, January, 2008).
"Squirt - Part VII: cue test machine results" (BD, February, 2008).
"Squirt - Part VIII: squerve effects" (BD, March, 2008).
"Squirt, swerve, and throw wrap-up" (BD, April, 2008).
Also, here's a video excerpt from Vol. II of the Video Encyclopedia of Pool Shots that explains and demonstrates things:
Now back to Diagram 3. Throw direction depends on the direction of the relative motion of the surface of the cue ball in contact with the object ball. This direction is affected by both cut angle and spin. "Throw - Part VI: inside/outside english" (BD, January, 2007) and "Throw - Part VII: CIT/SIT combo" (BD, February, 2007) illustrate the different possibilities quite well. The throw direction shown in Diagram 3 of "Squirt - Part I: introduction" (BD, August, 2007) is appropriate given the amount of english.
Object ball throw depends on cut angle, shot speed, type and amount of english, and the amount of vertical plane spin (draw, follow, stun). The following series of instructional articles elaborate on all of these factors:
"Throw - Part I: introduction" (BD, August, 2006).
"Throw - Part II: results" (BD, September, 2006).
"Throw - Part III: follow and draw effects" (BD, October, 2006).
"Throw - Part IV: spin-induced throw" (BD, November, 2006).
"Throw - Part V: SIT speed effects" (BD, December, 2006).
"Throw - Part VI: inside/outside english" (BD, January, 2007).
"Throw - Part VII: CIT/SIT combo" (BD, February, 2007).
"Throw - Part VIII: spin transfer" (BD, March, 2007).
"Throw - Part IX: spin transfer follow-up" (BD, April, 2007).
"Throw - Part X: the big picture" (BD, May, 2007).
"Throw - Part XI: everything you ever wanted to know about throw" (BD, June, 2007).
"Throw - Part XII: calibration, and hold shots" (BD, July, 2007).
Collision-induced throw (CIT) and spin-induced throw (SIT) are just different names for throw, depending upon the primary cause of the throw, but the effects don't really combine as separate factors.
straight-in shot with unintentional sidespin
What effects do squirt, swerve, and throw have with a straight-in shot hit with unintentional sidespin?
There are two possible cases here:
1.) The cue is aligned in the proper aiming line direction but shifted to the left a little, creating unintentionally left sidespin, but the stroke is straight. In this case, the CB will squirt to the right (the amount depends on the cue and the amount of tip offset), the CB will swerve back some to the left (the amount depends on shot speed, cue elevation and ball/cloth conditions), the contact point might be to the left or right of the initial target depending on the relative amounts of squirt and swerve, then the sidespin will throw the OB a little to the right of what the contact point suggests.
2.) The cue is aligned in the proper aiming line direction and the cue tip is aligned with the center of the CB, but the stroke is not perfectly straight, resulting in slight unintentional left sidespin. In this case, the aiming line is now pivoted to the left a little, so the CB will tend to head to the left a little (the amount will depend on bridge distance). Everything else is the same as with "1," but now relative to this new aiming line direction.
tip hardness effects
What differences does tip hardness make, and does it affect how much spin can be applied, or the amount of squirt that results?
See tip hardness effects.
tip contact height effects
What effect does tip contact height (for draw and follow) have on squirt or net cue ball deflection?
Hitting higher on the CB can do two important things related to net CB deflection (AKA squerve or the combined effects of squirt and swerve). Hitting higher can result in the cue being more level if the butt is lowered to help raise the tip. This would actually create less swerve, which would tend to exaggerate the effect of squirt (since less of the squirt is being cancelled by swerve). See squirt cue elevation effects for more info. However, with a higher hit on the ball, squirt actually has two components ... one sideways which causes CB deflection (what we normally call "squirt"), and one downward (into the table). The downward component will cause swerve to occur sooner (even before the CB moves forward very much at all). This is sometimes called "immediate swerve." This effect is more noticeable with highly-elevated-cue shots like massé shots and jump shots with off-center hits (intentional or not) that create a lot more swerve (CB curve) than with typical low-elevation pool shots. The immediate swerve associated with follow shots lessens the effect of sideways squirt (since more of the sideways squirt is being cancelled by the sooner swerve).
A draw shot, on the other hand, has less downward force into the table (from cue elevation) due to an upward component of squirt which reduces the "immediate swerve." Also, as illustrated in Diagram 1 of "Squirt - Part VIII: squerve effects" (BD, March, 2008), swerve takes longer to complete with a draw shot since the CB slides over a longer distance while the curving (swerve) takes place, before the CB heads in a straight line. Because the swerve occurs later, it doesn't cancel as much as the squirt effect, so the net or effective CB deflection will typically be larger with draw shots, especially with more speed (as long as the cue isn't elevated an extra amount, which causes more swerve). For more information, see squerve.
tip size and shape effects
Does the tip and shaft size and shape make a difference?
See tip size and shape effects.
what causes squirt?
What causes cue ball deflection (AKA "squirt")?
Check out the following article: "Squirt - Part I: introduction" (BD, August, 2007). It explains and illustrates what causes squirt in a very easy-to-understand way. Here's a simple explanation: With an off-center hit, while the tip is in contact with the CB, the CB starts to move forward and turn. The ball turn pushes the tip away sideways causing the end mass of the shaft to move. Mass doesn't like to move, so it pushes back during contact (because for every action, there is an equal and opposite reaction). That's why the CB deflects (squirts) off line.
In understanding the endmass effect, it helps to distinguish between the forward force (and impulse) required to deliver forward momentum to the cue and CB, and the sideways force (and impulse) resulting in endmass momentum and CB deflection. These are two different forces. The forward force and cue momentum is a result of what you develop and feel during your forward stroke into the ball. The sideways force (during the incredibly brief tip contact time) is a result of the interaction between the CB and endmass. This equal and opposite sideways force is what causes CB deflection and sideways momentum of the end mass, which in turn causes the cue flex and vibration (which you feel after the hit).
Here's a diagram (still images from a 2000 frame/sec high-speed video) and an explanation from the article, providing a more-detailed explanation for what causes squirt:
Still "a" is just before contact. Stills "b" through "e" represent a little less than 0.001 second (one thousandth of a second) during which the tip is in contact with the ball. In still "f" the tip hasn't fully recovered from the compression yet as the CB is separating. Still "g" is after separation. The line and arc appearing in each still mark the initial cue stick and CB positions. Notice how much the cue tip deflects away (down in the diagram) from its original line of action. Also notice how much the cue tip deforms (e.g., see still "d").
The black arrows in still "c" in the diagram illustrate the effect that causes squirt. While the tip is in contact with the ball, the ball starts rotating. This rotation (counterclockwise in the diagram) pushes the cup tip down a little during contact. Because the end of the shaft has mass (called endmass), it takes force to move the end of the shaft down as the ball rotates (because mass has inertia, and force is required to change its speed). And Isaac Newton said: "for every action, there is an equal and opposite reaction;" therefore, if the tip is being pushed down by the ball, the tip will push back with an equal and opposite force on the ball. This force is what causes squirt.
The amount of squirt (cue ball deflection) depends on the effective mass ("endmass") being deflected in the shaft. The "effective mass" depends on how far the tip deflection is "felt" down the length of the shaft as a sideways "wave" travels down the shaft toward the butt. Because the tip is in contact with the CB for such a short time, the wave does not travel very far (only about 5-10 inches). The distance it travels varies with shaft stiffness some. It travels faster (and longer) in a stiffer shaft involving more "effective mass" in the sideways deflection, which causes more squirt. However, it is the shaft endmass and not shaft flex that results in squirt. The following article provides some evidence to back up this claim: "Return of the squirt robot" (BD, August, 2008).
The cue tip continues to move sideways and eventually springs back and vibrates back and forth, but the CB is long gone by then, so the stiffness and spring-back of the shaft has no significant or direct influence on squirt.
For more information and relevant demonstrations, see:
low-deflection (LD) shafts
endmass and stiffness
squirt effects summary and demonstrations
cue tip contact time
cue vibration resource page
Does cue tip compression, tip hardness, and shaft flex affect CB deflection (squirt)?
Here are some example super-slow-motion videos showing how the tip deforms and how the shaft flexes during tip contact:
I know that when one looks at these videos, it is tempting to think that squirt (CB deflection) is caused completely by tip compression and shaft flex. However, it is best to ignore these effects when trying to understand the basics of squirt. Tip compression and shaft flex are really just side effects of the off-center-hit forces required to keep the tip from slipping on the CB.
Now, the more the tip compresses and flexes sideways, the longer the tip will tend to stay in contact with the CB. This would certainly result in more squirt (CB deflection) because effective "endmass" is larger with a longer contact time. Also, the more the tip flexes sideways, the more the endmass of the shaft moves sideways, which would also tend to create more squirt. A harder tip compresses and flexes less and results in a shorter tip contact time. Therefore, a harder would be expected to produce less squirt, assuming it is not heavier than the tip to which it is being compared (for more info, see cue tip hardness effects). However, the experiments documented in the Cue and Tip Testing for Cue Ball Deflection (Squirt) video seem to imply that tip type, hardness, and height have very little effect on squirt.
Shaft flex can also have an effect because it might cause some of the "endmass" to move faster than it would otherwise. This could contribute to more squirt, but I wouldn't expect this effect to be very significant.
Again, the main effect that causes squirt is: During the very brief moment while the cue tip is in contact with the CB during an off-center hit, the CB starts to turn. This pushes the cue tip sideways aways from the CB giving the end of the shaft some sideways speed. It takes force to do this since the end of the shaft has mass. For every action (sideways force pushing on the tip), there is an equal and opposite reaction (sideways force pushing back on the CB), causing the CB to squirt sideways with "deflection" off its expected path (i.e., the CB doesn't go straight).
The most effective way to reduce squirt is to reduce the effective "endmass" of the shaft (for more info, see low-squirt (low-cue-ball-deflection or LD) shafts). Keeping the tip contact time as short as possible (e.g., by using a harder tip) can also help.
Can the joint or butt of a cue affect CB deflection (squirt)?
Physics and careful testing clearly show that squirt (CB deflection) depends only on the effective endmass of the shaft. Therefore, the butt can have no effect on squirt (CB deflection). For more info, see what causes squirt (CB deflection).
However, for a given stroke speed and cue elevation, changing the joint or butt can have an effect on CB swerve and therefore net or effective CB deflection (AKA squerve). For example, if the weight of the butt is different, the CB speed will be different (for a given stroke). CB speed does not affect squirt, but it does affect swerve. Also, the butt and joint can affect the efficiency of the cue's hit, which can also change the CB speed and resulting swerve.
Swerve is a function of CB speed, cue elevation, and ball/cloth conditions. It has nothing to do with the properties of the shaft. That's why when shafts are tested for squirt (CB deflection), the machines should keep the cue horizontal so stroke speed and the butt will have no effect on swerve or the measurements of the shaft characteristics. Results from some squirt-testing machines (like Meucci's "Myth Creator" machine) can be misleading and seemingly in conflict with well understood and tested concepts. For more information, see the bullets on the squirt robot test results resource page.
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