Mechanism Mobility Calculation
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By varying Watt's original six-bar linkage, Mehdigholi and Akbarnejad created a 1 DOF walking mechanism that can be split into two sets: a 4-bar and 5-bar linkage. This separation is shown with two different highlights in the sketch below. Here, the four five bar linkage is composed of links 1, 4, 5, 3, and 3 ground but is not standard as it does not have two grounding points. In other words, it actually has two degrees of freedom, and requires a second system to restrict its position, velocity and acceleration. That role is fulfilled by conducting kinematic analysis on the fivefour-bar mechanism of links 1', 3', 2, and 6 (ground). Because of this setup, the conventional method of analyzing a Watt's Six-Bar Linkage will not work–this mechanism is lacking one more grounding point for this methodology. Instead, seeing that theta 2 and 3' are unknowns in the 54-bar linkage and that they can be found using two equations (one real, one imaginary), allows me to then solve for the unknowns in the "fourfive-bar" linkage of theta 4 and 5. (Note: All angles below are measured relative to the horizontal in a counterclockwise direction, as is set out by the convention established for this class.)
From this setup, there are two different vector loop equations:
In the analysis done below, I opted to make the equations less crowded by using Rx to represent the same as |Rx|. Also, along each step of the way, I solved equation 1 first to find restrictions pertinent to link 3 that can then be applied to the "four-bar" submechanism.
Separating equation 1 into real and imaginary then gives
Observing that as the grounded link, theta 6 is equal to 0, I now have
Squaring both equations above and adding them together allows for some simplification
Let a = R1', b = R3', c = R2, d = R6 (for additional simplifciation)
Dividing by 2ac:
Let K1 = d/a, k2 = d/c, k3 = (a2 - b2 + c2 + d2) / (2ac) and apply cos(x-y) = sin(x)sin(y) + cos(x)cos(y)
Finally, substitute the half-angle identities for sine and cosine
Now, since it is known that the linkage is an open four-bar linkage, I know:
Repeating the process above, but solving for theta 3 gives:
k4 = d/b and k5 = (c2 - a2 - b2 - d2) / (2ab) simplifies the equation above to:
Once more, the half-angle identities were used and the following equation for theta 3 was found
Conducting position analysis on the second equation was far more difficult than I anticipated. After attempting to simplify the following equations multiple times, I opted to use MATLAB's built in symbolic solver to find a long equation for theta 4 and theta 5.
MATLAB:
>> syms R11 t1 R33 t3 R5 t5 R4 t4;
>> eqns = [(R11*cos(t1))+(R33*cos(t3))-(R5*cos(t5))-(R4*cos(t4)) == 0, (R11*sin(t1))+
(R33*sin(t3))-(R5*sin(t5))-(R4*sin(t4)) == 0];
>> S = solve(eqns,[t5 t4]);
>> S.t5
ans =
-2*atan((4*R4*R11*tan(t1/2) - ((2*R4*R5 - 2*R11*R33 + R4^2 + R5^2 - R11^2 - R33^2 + R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 + 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2))*(2*R4*R5 + 2*R11*R33 - R4^2 - R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))^(1/2) + 4*R4*R33*tan(t3/2) + 4*R4*R11*tan(t1/2)*tan(t3/2)^2 + 4*R4*R33*tan(t1/2)^2*tan(t3/2))/(2*R5*R11 + 2*R5*R33 + 2*R11*R33 - R4^2 + R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2 + 2*R5*R11*tan(t3/2)^2 + 2*R5*R33*tan(t1/2)^2 - 2*R5*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)) - (4*(R4*R11*tan(t1/2) + R5*R11*tan(t1/2) + R4*R33*tan(t3/2) + R5*R33*tan(t3/2) + R4*R11*tan(t1/2)*tan(t3/2)^2 + R5*R11*tan(t1/2)*tan(t3/2)^2 + R4*R33*tan(t1/2)^2*tan(t3/2) + R5*R33*tan(t1/2)^2*tan(t3/2)))/(2*R5*R11 + 2*R5*R33 + 2*R11*R33 - R4^2 + R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2 + 2*R5*R11*tan(t3/2)^2 + 2*R5*R33*tan(t1/2)^2 - 2*R5*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))
2*atan((4*(R4*R11*tan(t1/2) + R5*R11*tan(t1/2) + R4*R33*tan(t3/2) + R5*R33*tan(t3/2) + R4*R11*tan(t1/2)*tan(t3/2)^2 + R5*R11*tan(t1/2)*tan(t3/2)^2 + R4*R33*tan(t1/2)^2*tan(t3/2) + R5*R33*tan(t1/2)^2*tan(t3/2)))/(2*R5*R11 + 2*R5*R33 + 2*R11*R33 - R4^2 + R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2 + 2*R5*R11*tan(t3/2)^2 + 2*R5*R33*tan(t1/2)^2 - 2*R5*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)) - (((2*R4*R5 - 2*R11*R33 + R4^2 + R5^2 - R11^2 - R33^2 + R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 + 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2))*(2*R4*R5 + 2*R11*R33 - R4^2 - R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))^(1/2) + 4*R4*R11*tan(t1/2) + 4*R4*R33*tan(t3/2) + 4*R4*R11*tan(t1/2)*tan(t3/2)^2 + 4*R4*R33*tan(t1/2)^2*tan(t3/2))/(2*R5*R11 + 2*R5*R33 + 2*R11*R33 - R4^2 + R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2 + 2*R5*R11*tan(t3/2)^2 + 2*R5*R33*tan(t1/2)^2 - 2*R5*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))
>> S.t4
ans =
2*atan((4*R4*R11*tan(t1/2) - ((2*R4*R5 - 2*R11*R33 + R4^2 + R5^2 - R11^2 - R33^2 + R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 + 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2))*(2*R4*R5 + 2*R11*R33 - R4^2 - R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))^(1/2) + 4*R4*R33*tan(t3/2) + 4*R4*R11*tan(t1/2)*tan(t3/2)^2 + 4*R4*R33*tan(t1/2)^2*tan(t3/2))/(2*R4*R11 + 2*R4*R33 + 2*R11*R33 + R4^2 - R5^2 + R11^2 + R33^2 + R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R4*R11*tan(t1/2)^2 + 2*R4*R11*tan(t3/2)^2 + 2*R4*R33*tan(t1/2)^2 - 2*R4*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R4*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R4*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))
2*atan((((2*R4*R5 - 2*R11*R33 + R4^2 + R5^2 - R11^2 - R33^2 + R4^2*tan(t1/2)^2 + R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R5^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 + 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2))*(2*R4*R5 + 2*R11*R33 - R4^2 - R5^2 + R11^2 + R33^2 - R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))^(1/2) + 4*R4*R11*tan(t1/2) + 4*R4*R33*tan(t3/2) + 4*R4*R11*tan(t1/2)*tan(t3/2)^2 + 4*R4*R33*tan(t1/2)^2*tan(t3/2))/(2*R4*R11 + 2*R4*R33 + 2*R11*R33 + R4^2 - R5^2 + R11^2 + R33^2 + R4^2*tan(t1/2)^2 - R5^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 - R5^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2*tan(t3/2)^2 - 2*R4*R11*tan(t1/2)^2 + 2*R4*R11*tan(t3/2)^2 + 2*R4*R33*tan(t1/2)^2 - 2*R4*R33*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 - 2*R11*R33*tan(t3/2)^2 - 2*R4*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R4*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2)))
Using trial and error to determine which function was valid, and then simplify() and collect(), I got the following two equations for theta 4 and 5:
T4 = 2*atan((((R4^2*tan(t1/2)^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R4^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R4*R5 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2 + R5^2*tan(t3/2)^2 + R5^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R11^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2) + 2*R11*R33*tan(t3/2)^2 - 2*R11*R33 - R33^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 - R33^2)*(2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R4^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R4*R5 - R5^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2 - R5^2*tan(t3/2)^2 - R5^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R11^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2) - 2*R11*R33*tan(t3/2)^2 + 2*R11*R33 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R33^2))^(1/2) + 4*R4*R11*tan(t1/2) + 4*R4*R33*tan(t3/2) + 4*R4*R11*tan(t1/2)*tan(t3/2)^2 + 4*R4*R33*tan(t1/2)^2*tan(t3/2))/(R4^2*tan(t1/2)^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R4^2 - 2*R4*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R4*R11*tan(t1/2)^2 + 2*R4*R11*tan(t3/2)^2 + 2*R4*R11 - 2*R4*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R33*tan(t1/2)^2 - 2*R4*R33*tan(t3/2)^2 + 2*R4*R33 - R5^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2 - R5^2*tan(t3/2)^2 - R5^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R11^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2) - 2*R11*R33*tan(t3/2)^2 + 2*R11*R33 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R33^2))
T5 = 2*atan((4*R5*R11*tan(t1/2) - ((R4^2*tan(t1/2)^2*tan(t3/2)^2 + R4^2*tan(t1/2)^2 + R4^2*tan(t3/2)^2 + R4^2 + 2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R4*R5 + R5^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2 + R5^2*tan(t3/2)^2 + R5^2 - R11^2*tan(t1/2)^2*tan(t3/2)^2 - R11^2*tan(t1/2)^2 - R11^2*tan(t3/2)^2 - R11^2 - 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R11*R33*tan(t1/2)^2 - 8*R11*R33*tan(t1/2)*tan(t3/2) + 2*R11*R33*tan(t3/2)^2 - 2*R11*R33 - R33^2*tan(t1/2)^2*tan(t3/2)^2 - R33^2*tan(t1/2)^2 - R33^2*tan(t3/2)^2 - R33^2)*(2*R4*R5*tan(t1/2)^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R4^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + 2*R4*R5*tan(t1/2)^2 + 2*R4*R5*tan(t3/2)^2 + 2*R4*R5 - R5^2*tan(t1/2)^2*tan(t3/2)^2 - R5^2*tan(t1/2)^2 - R5^2*tan(t3/2)^2 - R5^2 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R11^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2) - 2*R11*R33*tan(t3/2)^2 + 2*R11*R33 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R33^2))^(1/2) + 4*R5*R33*tan(t3/2) + 4*R5*R11*tan(t1/2)*tan(t3/2)^2 + 4*R5*R33*tan(t1/2)^2*tan(t3/2))/(R5^2*tan(t1/2)^2*tan(t3/2)^2 - R4^2*tan(t1/2)^2 - R4^2*tan(t3/2)^2 - R4^2 - R4^2*tan(t1/2)^2*tan(t3/2)^2 + R5^2*tan(t1/2)^2 + R5^2*tan(t3/2)^2 + R5^2 - 2*R5*R11*tan(t1/2)^2*tan(t3/2)^2 - 2*R5*R11*tan(t1/2)^2 + 2*R5*R11*tan(t3/2)^2 + 2*R5*R11 - 2*R5*R33*tan(t1/2)^2*tan(t3/2)^2 + 2*R5*R33*tan(t1/2)^2 - 2*R5*R33*tan(t3/2)^2 + 2*R5*R33 + R11^2*tan(t1/2)^2*tan(t3/2)^2 + R11^2*tan(t1/2)^2 + R11^2*tan(t3/2)^2 + R11^2 + 2*R11*R33*tan(t1/2)^2*tan(t3/2)^2 - 2*R11*R33*tan(t1/2)^2 + 8*R11*R33*tan(t1/2)*tan(t3/2) - 2*R11*R33*tan(t3/2)^2 + 2*R11*R33 + R33^2*tan(t1/2)^2*tan(t3/2)^2 + R33^2*tan(t1/2)^2 + R33^2*tan(t3/2)^2 + R33^2))
With the position analysis completed, the velocity and acceleration analysis was relatively straightforward.
Using omega 2 and 3 from above, I could again, solve for the second vector loop's velocity values
For the acceleration analysis, the previous vector loop equation for velocity was derived with respect to time a second time and algebraically manipulated for the desired values:
Once more, using the results of the first vector loop equation, the second was solved:
Using the relationships found above, the position, velocity, and acceleration of the "end effector"—the foot here or link 5—was plotted using MATLAB below. The input crank (Link 1 above) was set to rotate with a speed of 1.57 rad/s or 15 RPM clockwise. This full range of motion creates a crank-rocker six-bar linkage, as can be seen in the animation. Additionally, the animation features a ternary link, which was simply added to the link assembly in PMKS to illustrate the position of the "foot" as an offset to link 5. Since there is not a rotating slider, there is not a Coriolis acceleration, so instead the resolved acceleration at the third "joint" on link 5 is shown with the orange line in the attached PMKS animation. The position plot successfully resembles the desired trajectory for a "step" described by Mehdigoli and Akbarnejad in their paper. Additionally, some discontinuities are excluded in the velocity and acceleration plots to have a defined and reasonable plot.
