Chebyshev X Linkage (5 bars)
■ "Chebyshev-X" is X-bars honest mechanism, and Excellent light footwork apparatus.
This method is more basic than Klann linkage, or, Jansen Linkage. I think so.
cf. Chebyshev's Lambda Mechanism (wikipedia)
Tchebyshevs plantigrade machine -- 14 rigid bodies (why? , , too many)
My same functional implementation ( i.e. above system) --- 9 rigid bodies
cf. Klann linkage (wikipedia) [ including Jansen Linkage animation, too. ]
cf. Theo Jansen mechanism (YouTube)
-- 13 rigid bodies
My same functional implementation ( i.e. above system) --- 5 rigid bodies
cf. Leg mechanism/ Gallery [ 9 mechanisms are displayed.]
I'm not novice of Chebyshev-Linkage, I'm semi-professional. (I've studied for many years about this.)
In above picture;
BC // C'B' and,
△ABC ≡ △EC'B' , so, EB' // AC ---- i.e. parallel to horizontal body.
Very Good !!!
■ Is it needed the guide course for Node E? or, Is the heel needed? → No.
Just after critical ∠BAC or ∠C'EB' = 180° or 0°, Node E may bend to wrong side.
In general, from the continuity, it doesn't happen.
→ Especially, usually we implement the two vertical legs at before-end and back-end
of one horizontal brown bar, so such two feet make the same traces,
so if one foot is on the ground, another foot is also on the ground,
so horizontal bar keeps horizontal, so the continuity of brown bar is assured.
then, we don't need worry the wrong side bending bar problem.
[ Top view of horizontal bar ]
(back-end)◆------------●B'-----------◆M (before-end) : Layer 1
◆------------●B'1-----------◆W : Layer 2
[ In above picture, You can enjoy by sliding M or W ◆ position. ]
---------
① If necessary , Please set the guardrail to avoid the mistake.
(Node E trace course is the Chebyshev Linkage curve.)
At least, (-2,4) neighborhood local 2 guardrails are needed.
(-2,√24) critical E point neighborhood is, too.
→ I recommend the place near the horizontal bar both ends.
② Please append the heel to the foot-end. ---- i.e. ⊥ shape foot.
To keep EB' bar horizontal state, keep the continuity of the leg q vertical state.
★Tip. --- named "Chebyshev +X mix (6 bars)"
Please introduce blue bar, if you want (if possible). --- I don't recommend.
But Total number of bars becomes 6. +1 increases.
( It becomes more complex/ simple. (?) This 6 bars system is exaggerated and heavy. )
Even in this case, As a matter of fact, above critical problem (i.e wrong bending problem) remains yet.
But, in this case, above critical problem (i.e wrong bending problem) is resolved. And becomes robust.
[ Tip: C'EC'' bar is ------|------- style needed. A pair of pedals style. It makes to lose the simplicity. ]
■ Good property
Segment q (= MF) is vertical, so, if the ground is flat, F1F shoes fits to the ground.
Shoes bottom doesn't consume.
This foot can walk the beach.
■ 180° crank implementation is easy.
180° pedal is the purple ◆ point B'2.
CB'2 // JB, and CB'2 = JB.
So. Point B'1 makes 180° corresponding Chebyshev curve.
So, B'1P segment is horizontal.
Pedal B ● doubles as both 0° and 180°.
And, Especially, the same surface on the crank disc. --- this is great !!!.
■ Trace B' is Y-axis symmetry, why?
This is a good educational question. --- Feel by intuition.
∵ X-axis ⊥ Y-axis (i.e 90°). and, ∠BCB' is 90°.
So, ∠(X-axis and BC) = ∠(Y-axis and B'C)
and,
B'C = f(BC) --- B to B' mapping. (BC)2+(B'C)2 = constant.
So, if B is X-axis symmetry, then B' is Y-axis symmetry.
Y B to B' mapping by C mediator
| .B' 90° rotation (∠BCB' = 90°)
B･ | sum of distance square = const. [BB' = const, too.]
X------+ C (if BC increase, then B'C decrease.)
■ Summary
① We can simulate a human leg (right and left foot pair) walking by one motor and one crank.
Why needs the multi cranks?
② Fig2 (Green bars apparatus) is 5 bars system. But this is bad implementation system.
Why? ---- Please consider deeply.
(Bars and Nodes conflict happens.)
③ You can tune Node M ◆ and W ◆ position to avoid vertical line collision to B ●.
Please slide M or W , if necessary. As a result, we have long horizontal brown bars.
(We can extend the horizontal bar to the both before and back directions, and can have vertical line to the both ends.)
Vertical line collision to Node C ▲ line, Please lengthen the horizontal bar more.
In a nutshell, You should prepare the long horizontal bar to avoid the collision between Nodes and bars.
cf. Long bar picture. (Water_Vender, by Harunobu 1760)
[ If short shoulder pole, he can't walk. stumble. perhaps. ]
④ And, we can have 2N multiple layers linkage structure to the Node B ● one axis.
B A C
---|
|----▲ Black X (layer 1)
▲----| Purple X (layer 2)
J ～
|----▲ Black X (layer 2N-1)
▲----| Purple X (layer 2N)
---|
We can have -|----------|- style Crank Node A axis . This is good system.
In Fig2 structure, the number of layer per one B axis is only one.
We must have -|-|-|-|-|-|-|-|- style Crank Node A axis . This is bad system. complicated structure.
⑤ The system becomes near to the Klann linkage, or, Jansen Linkage.
■ Bar slide layer check:
Top View
B---A C'------C
B---------B' C'--E
E-----B'
Considerable Good !!
( From this view, Node C ▲ doesn't conflict vertical bar q. So, we can more short the horizontal bar. )