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Lecture 25: Modeling Diving I
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What do we need to do?
Figure out what and how to simplify
Build a physical model that we can work with
Once that is done, we know how to proceedalthough it may be difficult
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There are lots of models of the human body designed for multiple purposes
We are limited to rigid linksbut we can certainly replace the muscles and tendons with joint torques
Let’s restrict ourselves to planar dives
(Our experience so far with the bicycle suggests that three dimensionsmay mean trouble)
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Planar motion implies bilateral symmetry —treat both arms and both legs as single mechanisms
There are several diving models out there with varying numbers of links
I’m going to use an eleven link model of my own devising
C7
ankle
knee
hip
L1
T6elbow
wristshoulder
C1
The joints
very much not to scale
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head
foot
calf
thigh
pelvisabdomen
thorax
hand
lower armupper arm
Parts of the diver
neck
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I number the links in the following order
footlower legupper leg
pelvisabdomen
thoraxneckhead
upper armlower arm
hand
will be the reference link
I need a reference link because the whole mechanism is flying through the air
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diver is confined to the plane
all the parts of the diver (called links) are rigid
the centers of mass of the links are at their geometric centers
the diver can apply torques at all the joints
consider only the time between when the diver leaves the platform and when he/she hits the water
Formal assumptions
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Formulation
I have eleven links, hence 66 variables to start
I can confine the system to the x = 0 plane
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x i = 0, φi = π2
= θ i
f = π/2 moves the body x axis to the inertial y axisq = π/2 rotates the body z axis out of the plane and erects the body y axis
There are 33 of these, so the number of variables is now 33.
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There are no nonholonomic constraints
There are ten connectivity constraints relating the centers of mass
I consider the links to be connected along their J axes
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We have a choice to make at this point:apply the connectivity constraints or
transform them to pseudononholonomic constraints
Two dimensional connectivity constraints aren’t all that badand there are no nonholonomic constraints, so I’m applying them
There are twenty of these: the 2D centers of mass with respect to one reference link
This leaves me with thirteen variables: eleven ys and y5 and z5
(I am using link 5, the abdomen, as my reference link)
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Now it’s time to assign generalized coordinates
We could simply plug the variables into a q vectorbut I want to do something a little different
All the joint torques will act on two links at a timewith an equal and opposite torque
If I define qs as angle differences, this will be easier to apply
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So, my first three qs will be y, z and y of the reference link
The others will be angle differences
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q =
y5
z5
ψ 5
ψ 5 −ψ 4
ψ 4 −ψ 3
ψ 3 −ψ 2
ψ 2 −ψ1
ψ 6 −ψ 5
ψ 7 −ψ 6
ψ 8 −ψ 7
ψ 9 −ψ 6
ψ10 −ψ 9
ψ11 −ψ10
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⇔
abdomen - pelvispelvis − upper leg
upper leg - lower leglower leg - foot
abdomen - thoraxthorax - neckneck − head
thorax - upper armupper arm- lower arm
lower arm - hand
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There are no constraints, so the constraint matrix is null
There will be torques at all the joints, so nothing will be conserved
There’s no advantage in using a pure Hamilton approach in terms of p and q
We can use our normal approach once we realize that a null constraint matrixmeans that the null space matrix must span the entire space
We can use the 13 x 13 identity matrix as S and proceed from there.
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˙ W = Ti ⋅ω ii=1
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∑ ⇒ Qi = ∂ ˙ W ∂˙ q i
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˙ q i = S ji u j = ui
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˙ p i = ∂L∂qi + Qi = 1
2Mmn
∂qi umun − ∂V∂qi + Qi
and, as usual
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˙ p i = 12
M ij ˙ u j + 12
∂M ij
∂qk u juk
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Put all this together
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M ij ˙ u j = −∂M ij
∂qk u juk + 12
M jk
∂qi u juk − ∂V∂qi + ∂ ˙ W
∂˙ q i€
˙ q i = ui
We need to be a bit more precise about the rate of work termbut it is pretty easy because everything is two dimensional
We need to pick signs, assign a convention
Index each torque to correspond to its connection
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Start from the reference link and proceed distally in each direction
I will have the following torques
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τ 54,τ 43,τ 32,τ 21
τ 56,τ 67,τ 78
τ 69,τ 910,τ 1011
The first subscript is the “base” and the second the “recipient”
The general work term is
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τ ij˙ ψ j − ˙ ψ i( )
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This makes it clear why I chose the generalized coordinates I did
The generalized forces become
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Q = 0 0 0 −τ 54 −τ 43 −τ 32 −τ 21 τ 56 τ 67 τ 78 τ 69 τ 910 τ 1011{ }
one force per variable, which will be extremely helpful when we design our controlIt means that each Hamilton equation has at most one force in it
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We’re going to look at control another dayToday let’s just focus on the differential (and algebraic) equations
The q equations are pretty near trivial
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˙ q i = ui
I won’t worry about them for now
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€
M ij ˙ u j = −∂M ij
∂qk u juk + 12
M jk
∂qi u juk − ∂V∂qi + ∂ ˙ W
∂˙ q i
I want the equivalent equations using the method of Zs
We write
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pi = 12
M ij ˙ q j = 12
M ijSnjun = 1
2M iju
j = Zijuj
remember that S is the identity matrix
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M is pretty complicated in this formulation because we have applied all the constraints
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M =
• 0 • • • • • • • • • • •0 • • • • • • • • • • • •• • • • • • • • • • • • •• • • • • • • • 0 0 0 0 0• • • • • • • • 0 0 0 0 0• • • • • • • • 0 0 0 0 0• • • • • • • • 0 0 0 0 0• • • • 0 0 0 0 • • • • •• • • • 0 0 0 0 • • 0 0 0• • • • 0 0 0 0 • • 0 0 0• • • • 0 0 0 0 • 0 0 • •• • • • 0 0 0 0 • 0 0 • •• • • • 0 0 0 0 • 0 0 • •
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⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
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⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
There are 107 nonzero components
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This means 107 nonzero components of Z, each with its own algebraic equation
The gradient
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∂Zij
∂qk has 537 nonzero components (out of a possible 2197)
Operationally we simply charge ahead and calculate everything directly
I suppose we know how to find the Lagrangian
how to apply holonomic constraintshow to assign generalized coordinates
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The momentum
The Zs
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Gradients of the Zs
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Those were the actual ones; we need symbolic equivalents.
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So we have 26 differential equations (thirteen for q and thirteen for u)and 642 accompanying algebraic equations
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We can see all of this in context in Mathematicabut there is nothing to run until we understand the torques
and that’s going to wait until next time