Robotics Fundamentals 1 gave the overall picture and put structure first in the list of subsystems. This chapter looks at it properly. The structure of a robot holds the motors, sensors, board and battery in the right places, carries their loads without bending too much, and decides whether the robot stays upright. Most first robots that fail do so for mechanical reasons: a frame that flexes so the wheels rub, a battery that puts the weight in the wrong place so the robot tips backwards on a ramp, a screw that strips out of a printed part. All of it can be checked with arithmetic before anything is cut or printed.
The material values below (stiffness, density, yield strength) and the fastener figures are typical, approximate values from memory. Real materials vary by grade, supplier and, for printed parts, by print settings, so use the numbers to compare options and to check the order of size, and read the datasheet before relying on one. The formulas themselves are standard.
1. Frames and materials
A frame has three jobs: locate parts (motors in line, sensors facing the right way), carry loads (weight, motor thrust, a knock against a wall) and stay stiff enough that parts do not move relative to each other. Weight matters too: every gram has to be carried by the motors and paid for by the battery (Basic Electronics 11 gave the energy side of that). The design question is always how much stiffness and strength you get for each gram.
Typical materials
Material
Stiffness E (GPa)
Density (kg/m³)
Yield or strength (MPa)
E / density (MN·m/kg)
In robots
Aluminium alloy (6061-T6)
69
2700
about 275 (yield)
25.6
Plates, angle, tube, arms and brackets. Light, easy to cut and drill, does not rust.
Mild steel
200
7850
about 250 (yield)
25.5
Shafts, big frames, anything that must be cheap and stiff and weight does not matter. Rusts unless coated.
PLA (3D printed)
about 3.5
about 1250
about 50, lower between layers
2.8
The easy plastic to print. Stiff but brittle, and softens at around 55 to 60 °C, so keep it away from hot motors and cars in summer.
ABS or PETG (printed)
about 2 to 2.5
about 1050 to 1270
about 40 to 50
about 2 (ABS)
Tougher and more heat-resistant than PLA; ABS is harder to print, PETG easier.
Acrylic (cast PMMA sheet)
about 3
about 1190
about 70, brittle
2.5
Laser-cut chassis plates. Cracks from sharp corners and over-tight screws.
Birch plywood
about 9 along the face grain
about 680
varies with grade and direction
13.2
Laser-cut or hand-cut plates; cheap, forgiving, good stiffness for its weight, best kept dry.
Carbon-fibre laminate
about 60 (plate with fibres in several directions); about 130 along the fibres of a unidirectional layer
about 1600
several hundred, brittle
about 37.5 (plate)
Tubes and plates where low weight is worth the cost (arms, drones). Conducts electricity: keep it away from bare wiring.
The last numeric column, E / density, is the stiffness you get per kilogram of material. Notice that aluminium and steel are almost identical on it, even though steel is three times heavier: for a given mass you can have a steel part or an aluminium part about three times as thick, and thickness is what stiffness depends on most (next section). Plastics are about ten times worse than metals per kilogram, so a plastic part must be made thicker or ribbed to match. Carbon fibre is the one that beats the metals.
Change the shape before you change the material
How a part is shaped usually matters more than what it is made of. A beam's bending stiffness depends on E and on the shape through I, and I grows with the cube of the depth. Making a flat strip stand on edge, adding a rib or folding a flange gives a gain that switching material rarely can. The example below shows a factor of 44 from turning a bar on its side.
Beam bending: the cantilever
Most robot structure is some form of cantilever: a beam fixed at one end and loaded at the other (an arm held out from a motor, a sensor mast, a bracket holding a motor). Two things go wrong with it. It can deflect too much (the sensor points at the floor, the arm droops), or it can yield (bend permanently or break). The formulas give both:
Cantilever, length L, load F at the free end, material stiffness E, second moment of area I:
Tip deflection: delta = F L^3 / (3 E I)
Rectangular section: I = b h^3 / 12 (b = width, h = depth measured IN THE DIRECTION OF THE LOAD)
Bending moment at the wall: M = F L
Greatest stress (outer face at the wall): sigma = M c / I = (F L)(h / 2) / I = 6 F L / (b h^2)
Safety factor: SF = yield strength / sigma
Cross-section orientation (the same bar, 20 mm x 3 mm; load acts downwards):
FLAT (load across the 3 mm) ON EDGE (load along the 20 mm)
+--------------------+ +---+
| | h = 3 mm | |
+--------------------+ | |
b = 20 mm | | h = 20 mm
| |
I = 20 x 3^3 / 12 = 45 mm^4 | |
+---+
b = 3 mm
I = 3 x 20^3 / 12 = 2000 mm^4
Beam sketch: ||=============================+
|| fixed length L |
|| v F (the free end drops by delta)
Two scalings are worth remembering. Deflection grows with the cube of the length (twice as long: eight times the droop) and falls with the cube of the depth (twice as deep: one eighth). Stress grows with the length but falls with the square of the depth.
Worked example: a sensor arm made from 6061 aluminium flat bar, b = 20 mm, h = 3 mm, L = 0.15 m, carrying F = 5 N (about 510 g) at the tip.
Use E = 69 GPa and yield = 275 MPa (typical values).
Flat (h = 3 mm):
I = 0.020 x 0.003^3 / 12 = 4.5e-11 m^4
delta = F L^3 / (3 E I) = 5 x 0.003375 / (3 x 69e9 x 4.5e-11) = 0.016875 / 9.315 = 0.00181 m = 1.81 mm
sigma = 6 F L / (b h^2) = 6 x 5 x 0.15 / (0.020 x 0.003^2) = 4.5 / 1.8e-7 = 25 MPa
SF = 275 / 25 = 11 (very safe against yielding, yet the tip still droops 1.8 mm: STIFFNESS is the limit, not strength)
The same bar turned on edge (b = 3 mm, h = 20 mm):
I = 0.003 x 0.020^3 / 12 = 2.0e-9 m^4 (44.4 times larger, because (20/3)^2 = 44.4)
delta = 0.016875 / (3 x 69e9 x 2.0e-9) = 0.0000408 m = 0.041 mm
sigma = 6 x 5 x 0.15 / (0.003 x 0.020^2) = 4.5 / 1.2e-6 = 3.75 MPa SF = 73
The same flat bar twice as long (L = 0.30 m): delta = 1.81 x 8 = 14.5 mm, sigma = 50 MPa, SF = 5.5
The same flat bar, printed in PLA instead (E = 3.5 GPa): delta = 1.81 x (69 / 3.5) = 35.7 mm (20 times the droop; stress is unchanged at 25 MPa,
which against PLA's roughly 50 MPa leaves SF of only about 2, and less across the layers)
Mass of the flat bar: aluminium 0.020 x 0.003 x 0.15 x 2700 = 24.3 g; PLA 0.020 x 0.003 x 0.15 x 1250 = 11.3 g
Safety factors are not decoration
The formulas give the stress in an ideal beam under a known, steady load. Real robots get dropped, driven into walls and loaded at the corner of a bracket where stress concentrates. A common rule of thumb is a safety factor of about 2 to 3 for steady loads on ductile metal, and 5 or more for brittle materials (acrylic, carbon fibre, PLA) or for loads that arrive as knocks. Check also what gives way first: it is usually a joint, a bolt hole or a thin printed neck, not the long bar.
2. Levers, linkages and joints
Moment (torque) balance
The turning effect of a force is its moment, also called torque: the force multiplied by the perpendicular distance from the pivot to the line of the force. A body that is not turning has its moments balanced, and one that is not moving has its forces balanced. These two conditions solve most mechanics problems on a robot.
Moment of a force about a pivot: M = F x d (d = perpendicular distance, in metres; result in N m)
Balance (equilibrium): sum of moments about any point = 0 sum of forces = 0
Lever mechanical advantage: MA = d(effort) / d(load) F(effort) = F(load) / MA
Simple lever (effort and load on opposite sides of the pivot):
d(effort) = 0.30 m d(load) = 0.05 m
effort | | load (a weight W)
v | v
========O=================================================^== (O is the pivot)
The trade: a lever gives a larger force at the cost of a smaller movement. The effort moves MA times farther than the load.
Worked example 1: a 2 kg load hangs 0.05 m from the pivot of a lever, and you push 0.30 m from the pivot.
W = m g = 2 x 9.81 = 19.62 N MA = 0.30 / 0.05 = 6 F(effort) = 19.62 / 6 = 3.27 N
Worked example 2: a hobby servo (stall torque about 1.2 N m, taken as a typical figure) holds out an arm of length L = 0.20 m. The arm weighs 0.05 kg,
with its weight acting at its middle (0.10 m).
Torque from the arm itself: 0.05 x 9.81 x 0.10 = 0.049 N m
Torque left for a load at the tip: 1.2 - 0.049 = 1.151 N m
Load force at the tip: F = 1.151 / 0.20 = 5.76 N m = 5.76 / 9.81 = 0.587 kg (at the very limit of the servo)
A servo should not run at its stall torque for long, so a design should keep well below it (Robotics Fundamentals 9 gives joint torque in detail).
Worked example 3: play in a joint grows with the length of what is attached to it.
0.5 degrees of free play in a joint, measured at the end of a 0.20 m arm: 0.20 x tan(0.5 deg) = 0.00175 m = 1.7 mm of wobble at the tip.
Linkages
A linkage is a set of rigid links joined by joints so that one input motion produces a controlled output motion. Gruebler's formula from Robotics Fundamentals 1 counts the freedom: the four-bar linkage has one, so one motor drives it. The four-bar is the workhorse: it appears in grippers (a servo opens two jaws), in parallelogram arms that keep a platform level while it lifts, and in the legs of walking robots.
Four-bar linkage: ground link (fixed), crank (driven by the motor), coupler, rocker (the output)
coupler
B o===============o C
/ \
crank / \ rocker
/ \
A o=======================o D A and D are pivots on the ground link
ground
Grashof's rule: let s = the shortest link, l = the longest, p and q = the other two.
If s + l <= p + q, then the shortest link can turn through a full circle relative to the others.
If the shortest link is next to the ground (a driven crank), the result is a crank-rocker: a motor turning round and round makes the rocker swing to and fro.
Worked example: ground 90 mm, crank 40 mm, coupler 100 mm, rocker 80 mm. s = 40, l = 100, p + q = 90 + 80 = 170; s + l = 140 <= 170: yes.
The 40 mm crank is the shortest link and is next to the ground, so it can turn continuously and the 80 mm rocker swings.
Mechanical advantage of a linkage changes through the motion. Near the end of a stroke, when the crank and coupler line up, a small input force
gives a very large output force (a toggle). A gripper uses this to clamp firmly without the servo working hard.
Joints
Joint
Allows
Typical construction
Watch for
Revolute (hinge)
Rotation about one axis
A shoulder screw or pin through the links, a bushing, a ball bearing, or the output shaft of a servo or motor
Play (slop) and friction: a loose pin gives wobble at the tip, an over-tight one binds
Prismatic (slider)
Sliding along one axis
Linear rail and carriage, round rods in bushings, a lead screw
Binding if the two rods are not parallel; dirt on the rails
Ball joint
Rotation about three axes
Ball-and-socket, used on pushrods and link ends
Slack, and it adds freedom you may not want
Flexure (living hinge)
A small bend, from a thin section
A thin band left in a printed or cut part
Fatigue: it cracks after many bends in a brittle plastic
Fixed (rigid)
Nothing
Screws, glue, a clamp
A joint you meant to be rigid but that flexes is the usual cause of a floppy frame
Wherever a joint turns, friction costs torque and play costs accuracy. A ball bearing is a bit more costly than a plain bushing, but it turns with very little friction and little play. Where a robot has to repeat a motion accurately (an arm picking up a block), put the effort into rigid links and tight, well-supported joints before a better motor.
3. Centre of gravity and stability
All the weight of a robot can be treated as acting at a single point, its centre of gravity (CoG). Where that point is decides whether the robot stays on its wheels. For a robot made of several parts, find it by taking moments: it is the weighted average of the positions of the parts, with mass as the weight.
Centre of gravity of parts i = 1, 2, 3 ... with masses m(i) at positions x(i) (along the robot) and z(i) (height above the ground):
M = sum of m(i) x(cg) = sum of m(i) x(i) / M z(cg) = sum of m(i) z(i) / M
(the same idea works for y, across the robot)
Worked example: the small two-wheeled robot, x measured forward from the drive axle, z up from the floor, all in metres.
The caster (a free-swivelling ball or wheel at the front) touches the ground at x = +0.090.
Part m (kg) x (m) z (m) m x m z
Chassis plate 0.20 0.000 0.030 0.00000 0.00600
Two motors and gearboxes 0.12 0.000 0.032 0.00000 0.00384
Two wheels 0.06 0.000 0.032 0.00000 0.00192
Battery pack 0.25 -0.040 0.045 -0.01000 0.01125
Controller board 0.05 0.020 0.060 0.00100 0.00300
Caster 0.03 0.090 0.012 0.00270 0.00036
Front sensor board 0.02 0.090 0.020 0.00180 0.00040
Total 0.73 -0.00450 0.02677
x(cg) = -0.00450 / 0.73 = -0.00616 m (6 mm BEHIND the drive axle)
z(cg) = 0.02677 / 0.73 = 0.03667 m (36.7 mm above the floor)
The support polygon
A robot is stable while the CoG, projected straight down onto the floor, lies inside the support polygon: the shape formed by joining the points where the robot touches the ground. Two wheels and a caster make a triangle; four wheels make a rectangle. The result for the example is bad news:
Top view (the floor). Forward is up the page.
C caster contact (x = +0.090 m)
/|\
/ | \
/ | \
/ | \
/ | \
x = 0 L-----+-----R wheel contacts (x = 0, y = -0.060 and +0.060 m)
* CoG, as first calculated (x = -0.006 m: just OUTSIDE the triangle, behind the wheel line)
The CoG is behind the axle line. The robot will sit on its tail and the caster will hang in the air, so it cannot be driven as designed.
The fix: move the battery forward. What battery position puts the CoG at x = +0.020 m (inside the triangle, with some weight on the caster)?
Required total moment: 0.020 x 0.73 = 0.0146. Moment of everything but the battery: 0.00100 + 0.00270 + 0.00180 = 0.00550.
Battery moment needed: 0.0146 - 0.0055 = 0.0091 so x(battery) = 0.0091 / 0.25 = +0.0364 m (36 mm FORWARD of the axle)
Check: x(cg) = (0.0055 + 0.25 x 0.0364) / 0.73 = 0.0146 / 0.73 = +0.0200 m. Weight W = 0.73 x 9.81 = 7.16 N.
Where the weight is carried (moments about the axle): caster share = x(cg) / 0.090 = 0.020 / 0.090 = 22.2 % = 1.59 N; the two wheels carry the other 5.57 N.
(Too little on the caster and it loses contact over bumps; too much and the drive wheels lose grip. A fifth to a third on the caster is typical.)
Tipping on a slope, and when it tips instead of slipping
A robot tips over an edge of its support polygon when the line of its CoG, which stays vertical, crosses that edge. On a slope, the line of weight stays vertical while the robot tilts, so it moves relative to the contact points. At the tipping angle, the CoG is directly above the tipping edge. With h the height of the CoG above the ground and d its horizontal distance from the edge it will tip over:
Tipping angle on a slope: tan(theta) = d / h d = horizontal distance from the CoG to the tipping edge
Sideways, across a slope: d = track / 2 (CoG centred between the wheels): tan(theta) = (track / 2) / h(cog)
Wheels slip instead when: tan(theta) = mu (mu = friction coefficient between the tyre and the surface)
The robot tips FIRST if d / h < mu and slips first if mu < d / h.
Tipping or skidding in a straight line or a turn (acceleration a on level ground, the same logic): it tips when a = g d / h.
Worked example 1: the corrected small robot (track 0.12 m, h = 0.0367 m, x(cg) = +0.020 m, caster at +0.090 m, rubber tyres on a dry surface, mu about 0.8).
Sideways: tan(theta) = 0.060 / 0.0367 = 1.636 theta = 58.6 deg
Facing uphill: it tips backwards about the wheel axle: d = 0.020 tan(theta) = 0.020 / 0.0367 = 0.545 theta = 28.6 deg
Facing downhill: it tips forwards about the caster: d = 0.090 - 0.020 = 0.070 tan(theta) = 1.91 theta = 62.4 deg
Wheels slip at: tan(theta) = 0.8 theta = 38.7 deg
So uphill the robot tips backwards at 28.6 deg, BEFORE it would slip (38.7 deg). The robot with a low, short body is stable in every direction but one: it needs the
CoG moved forward, or lowered, to climb steeply. Moving the CoG 10 mm forward adds: tan(theta) = 0.030 / 0.0367 = 0.817, theta = 39.3 deg.
Worked example 2: a tall robot (track 0.15 m, CoG height 0.15 m).
Sideways tipping: tan(theta) = 0.075 / 0.15 = 0.5 theta = 26.6 deg (a ramp with a 1 in 2 slope across it)
Tips in a turn at a(lateral) = g x (track / 2) / h = 9.81 x 0.5 = 4.9 m/s^2. A turn of radius 0.5 m has a = v^2 / r, so v = sqrt(4.9 x 0.5) = 1.57 m/s.
Braking: if the CoG is 0.10 m behind the front contact line, a robot tips forwards at a = 9.81 x 0.10 / 0.15 = 6.5 m/s^2 (the same rule).
Four ways to make a robot more stable
Lower the CoG (heavy parts, mainly the battery, low in the frame), widen the track and wheelbase (a bigger polygon), centre the CoG in the polygon, and slow down or smooth the acceleration (tipping in a turn depends on speed squared). Put the heaviest parts low and between the wheels. A tall mast with a camera or sensor on top raises the CoG a great deal: weigh it as part of the sum above, not as an afterthought.
4. 3D-printed and laser-cut parts
Making your own parts is the great strength of hobby robotics: a plate with exactly the right motor holes, drawn in an afternoon and made in an hour. The two common methods have different strengths. The design rules below are typical values from memory; the limits of your own machine, and of the material you use, are what count.
3D printing (filament)
Laser cutting (sheet)
Shape
Almost any 3D shape: brackets, enclosures, wheels, gears
Flat parts, cut out of 3 to 6 mm sheet; 3D shapes by slotting parts together
Materials
PLA, PETG, ABS, nylon, flexible TPU
Acrylic, plywood, MDF, card. Never PVC (it gives off chlorine)
Strength
Directional: weakest between the layers
The sheet is the same in all directions across its plane
Accuracy
About 0.2 mm; holes print undersized
About 0.1 mm, with the kerf (width of the cut) to allow for
Speed and cost
Slow (hours for a big part), cheap in material
Fast (minutes), cheap for flat parts
Good for
One-off complex shapes, mounts, covers
Chassis plates, decks, a quick first frame
Design rules for printed parts
Print so the load runs along the layers, not across them. A printed part is much weaker where one layer meets the next, often only about half to two thirds of the strength within a layer. A bracket lying flat is stronger in bending than the same one standing on its edge when the load pulls the layers apart.
Thickness is cheap stiffness. Doubling the thickness of a plate makes it 8 times stiffer (the h³ rule above) for twice the plastic. Ribs and fillets do the job for less. Round the inside corners of cut-outs to avoid cracks starting there.
Walls and infill: a few perimeters (2 to 4) and 20 to 40 % infill is a common default; the walls carry most of the load, not the infill.
Holes print small, by about 0.2 mm. Print a hole for an M3 screw at 3.4 to 3.6 mm, not 3.0 mm, and check with a test piece. Slots and press fits need testing the same way.
Overhangs steeper than about 45 degrees from the vertical need support or they sag; design a part so that it can print without support where it can.
Heat: PLA softens at around 55 to 60 °C. Motors under load, drivers and sunlit dashboards can reach it; use PETG or ABS there.
Design rules for laser-cut parts
Kerf. The laser removes a strip about 0.1 to 0.2 mm wide. A slot cut to exactly the sheet thickness is a loose fit; draw it a hair narrower, or adjust the drawing by the machine's kerf. Test on scrap.
Tab and slot joints. Parts slot together at right angles and are held with a screw and nut in a captured slot (a nut trap). That gives a stiff box from flat sheet.
Keep holes clear of edges (at least the hole's diameter, better twice) and round the corners, particularly in acrylic.
Acrylic cracks. Do not over-tighten screws, and use washers.
5. Fasteners
The robot's parts are held together mostly by metric screws. Their sizes are written M2, M3, M4 and so on: the M means metric and the number is the nominal diameter in millimetres. M3 (3 mm) is the commonest size on small robots. A screw is also described by its length (under the head for most types), its head (cap head with a hex socket, button head, countersunk) and its property class (8.8 is a common steel grade for general use; stainless steel A2-70 is common too).
Item
What it does
Note
Machine screw and nut
Clamps parts together through clearance holes
The normal choice for metal and acrylic; a washer under the head and under the nut spreads the load on plastic
Nyloc nut
A nut with a nylon ring that grips the thread
Resists vibration without a threadlocker; do not reuse many times
Standoff (spacer)
Holds two plates apart at a fixed distance, or mounts a board off a plate
Threaded hex standoffs in M3 are common; a Pico-class board uses M2 screws through holes of about 2.1 mm
Heat-set insert
A brass threaded insert pushed into a printed hole with a hot iron
Gives a strong metal thread in a plastic part that can be unscrewed many times
Self-tapping screw
Cuts its own thread in plastic or thin sheet
Fine for light, rarely removed parts; strips out when overused
Threadlocker
Liquid that hardens in the thread and stops it vibrating loose
Blue is medium-strength (can be undone with tools); red is permanent. Use on metal threads only: it can attack some plastics
Clearance holes and tolerances
A tolerance is the allowed variation in a size. A screw has to pass through a hole that is a little bigger than it is, so that parts can be assembled even if holes are slightly out of position. The standard clearance diameters for metric screws (from memory of the ISO 273 table) are:
Screw Close (fine) hole Normal (medium) hole Tap-drill (for cutting a thread) Printed hole (aim for)
M2 2.2 mm 2.4 mm 1.6 mm about 2.4 to 2.6 mm
M3 3.2 mm 3.4 mm 2.5 mm about 3.4 to 3.6 mm
M4 4.3 mm 4.5 mm 3.3 mm about 4.5 to 4.7 mm
Use the normal hole by default. Use the close hole where two parts must locate accurately. For heat-set inserts, the hole size depends on the insert: use the datasheet's figure.
Stack-up: tolerances ADD. Two parts joined by a screw, each with a hole position out by 0.1 mm, can be out by 0.2 mm in total, so the clearance has to be at least that
much bigger than the screw. Slotted holes let you adjust a motor mount's position.
Screws: shear, preload and tightening (approximate)
Three simple estimates tell you whether a bolted joint is sensible. These are rough guides: the real values depend on the grade, the thread, lubrication and the materials joined, and the margin you leave should reflect that.
Tensile stress area (the effective area of the threaded part): A(t) = (pi / 4)(d - 0.9382 p)^2 d = nominal diameter, p = thread pitch
Shear strength (approximate): about 0.6 x ultimate tensile strength. Class 8.8: ultimate about 800 MPa, proof about 580 MPa (typical figures)
Single-shear capacity: F(shear) = 0.6 x 800 x A(t)
Preload (the clamping force from tightening), aimed at about 70 % of the proof load: F(pre) = 0.7 x 580 x A(t)
Tightening torque (approximate): T = K F(pre) d K about 0.2 for a dry thread (the scatter is large, +/- 25 % or more)
Bearing stress on a plastic wall: F / (d t) d = hole diameter, t = wall thickness
Screw pitch p A(t) F(shear), 8.8 F(pre) T
M2 0.4 mm 2.07 mm^2 995 N 842 N 0.34 N m
M3 0.5 mm 5.03 mm^2 2415 N 2043 N 1.23 N m
M4 0.7 mm 8.78 mm^2 4214 N 3564 N 2.85 N m
Worked example: A(t) for M3 = (pi / 4)(3 - 0.9382 x 0.5)^2 = (pi / 4)(2.5309)^2 = 5.03 mm^2; F(shear) = 0.6 x 800 x 5.03 = 2415 N (about 246 kg);
F(pre) = 0.7 x 580 x 5.03 = 2043 N; T = 0.2 x 2043 x 0.003 = 1.23 N m.
Now a motor bracket in 3 mm printed PLA, held by M3 screws through 3 mm holes. Say the plastic can take about 20 MPa in bearing (a rough figure).
F(bearing) = 20 MPa x (3 mm x 3 mm) = 20 x 9 = 180 N per screw
The screw itself takes 2415 N in shear: 13 times more. THE PLASTIC FAILS FIRST, not the screw.
So in plastic you do not tighten to 1.2 N m: the plastic would crush or the thread strip long before. Tighten until snug, use a washer, and for threads that will be
undone, use a heat-set insert. The 1.23 N m figure is for steel screws in metal.
Thread engagement
A screw that is screwed only a turn or two into a threaded hole can strip. As a rule of thumb the engagement should be at least 1 to 1.5 times the screw diameter in steel and about 1.5 to 2 times in aluminium; plastic threads are weak, so use an insert. Leave the screw ends slightly short of the other side of the hole, or check that they do not hit a wire.
6. Challenges
Challenge 1: Deflection, stress and shape
A sensor arm is a 6061 aluminium flat bar, 25 mm wide and 4 mm thick, 0.20 m long, fixed at one end and carrying a load of F = 8 N at its tip (E = 69 GPa, yield 275 MPa). (a) Mounted flat (4 mm in the direction of the load), find I, the tip deflection, the greatest stress and the safety factor. (b) Turned on edge (25 mm in the direction of the load), find the same quantities, and the factor by which the deflection falls. (c) The flat bar is lengthened to 0.30 m: find the new deflection, stress and safety factor. (d) The flat 0.20 m bar is replaced by the same section in ABS (E = 2.2 GPa): by what factor does the deflection grow? (e) For the flat 0.20 m bar, find the largest tip load that keeps the deflection under 1 mm.
A small robot has these parts (x forward from the drive axle, z above the floor): chassis 0.25 kg at x = 0, z = 0.030 m; motors 0.14 kg at 0, 0.032; wheels 0.08 kg at 0, 0.033; battery 0.30 kg at x = −0.020, z = 0.040; controller 0.06 kg at 0.030, 0.065; sensor mast 0.05 kg at 0.120, 0.090; caster 0.03 kg at 0.110, 0.010. The caster touches the ground at x = 0.110 m and the track (distance between the wheels) is 0.14 m. (a) Find the total mass and the CoG (x and z). Is the CoG inside the support polygon, and what share of the weight does the caster carry? (b) Find the tipping angle sideways, and the tipping angle facing uphill and downhill. The tyres have mu = 0.8: which happens first facing uphill, tipping or slipping? (c) Find the acceleration at which the robot lifts its front (a = g x(cg) / h) when accelerating forward. (d) Where must the battery go (same height) for the CoG to be at x = +0.030 m? Find the new uphill tipping angle and the new caster share. (e) Name two other ways to improve the uphill stability.
A servo with a stall torque of 1.5 N m holds out an arm 0.25 m long. The arm weighs 0.08 kg (weight acting at its middle) and carries a gripper of 0.05 kg at the tip. To avoid running near stall, the design may use only 60 % of the stall torque. (a) Find the torque the servo needs just to hold the arm and gripper horizontal. (b) Find the heaviest payload (in kg) the gripper may carry at the tip. (c) The arm joint has 0.7 degrees of free play. Find the wobble at the tip. (d) An M3 class 8.8 screw (pitch 0.5 mm) holds the arm bracket. Find its tensile stress area, its single-shear capacity (0.6 x 800 MPa), its preload (70 % of 580 MPa) and the approximate tightening torque (K = 0.2). (e) The bracket is 3 mm printed PLA with a 3 mm hole, and the plastic can bear about 20 MPa. Find the bearing capacity of one screw hole, compare it with the screw's shear capacity, and say which fails first and how the bracket should be assembled.
Materials: aluminium and steel have about the same E/density; carbon fibre is better, plastics about ten times worse; change the shape before the material
Cantilever: deflection = F L³ / (3 E I); rectangle I = b h³ / 12 (h along the load); stress = 6 F L / (b h²); safety factor = yield / stress
Scaling: deflection ∝ L³ and 1/h³; stress ∝ L and 1/h²; a bar on edge is (h/b)² times stiffer than flat
Moments: M = F d; balance: the sum of moments = 0; lever MA = d(effort) / d(load); play at the tip = L tan(angle)
Linkages: four-bar has 1 DOF; Grashof: s + l ≤ p + q lets the shortest link turn fully
CoG: x(cg) = Σ m x / Σ m, z(cg) = Σ m z / Σ m; stable while the CoG is over the support polygon
Tipping: tan(theta) = d / h (sideways d = track/2); slips first if mu < d/h; tips at a = g d / h
Parts: load along the printed layers; add about 0.2 mm to printed holes; allow for laser kerf; PLA softens near 55 to 60 °C
Screws: A(t) = (π/4)(d − 0.9382 p)²; shear about 0.6 × UTS × A(t); preload about 0.7 × proof × A(t); T = 0.2 F d; in plastic the plastic fails first, so use washers and inserts
Next, Robotics Fundamentals 3 turns to motion: how a motor's torque, speed and power are chosen, how gears and gearboxes trade speed for torque, how belts, chains and wheels deliver it to the ground, and how to size a motor for the robot you have just designed the structure of.