MATH 447 Project 1:

Kinematics of the Stewart Platform

Omar Safsaf, Sophaktra Suy, Tan Tran

For this project, we will be focusing on the kinematics of a two-dimensional Stewart platform. Specifically, we will be looking at the possible poses of the platform given the dimensions of the triangle. In addition, we are given the struts’ lengths and anchor points. To find out the position of the platform, we needed to solve for θ, x, y given the prior information. To find the angle θ, we solve the equation given in our MATLAB function “Reality1”. To find the roots of the function, we analyze the plot of the function to find intervals we need to pass into our bisection method. We first apply this given the dimensions of the platform L₁=2, L₂=L₃=2^1/2, the strut lengths p₁= p₂= p₃=5^1/2. When we plot this, we are given the graph:

We know that the solution is zero at ±π/4.To make sure our plot is correct, we use our “bisect1”code on the intervals [-2,0] and [0,2] on our function and find that the roots returned are ±0.7854 or ±π/4. Next, we plot one of the poses of the platform:

Next, we solve the forward kinematics problem where x₁ = 5,(x₂,y₂) = (0,6),L₁ = L₃ = 3,L₂ = 3*2^1/2,γ = π/4,p₁ = p₂ = 5,p₃ = 3. When we run this through our program, “Run2”, we are given the graph of the function:

which we use to give our bisection method the intervals [-1,-0.5], [-0.5,0], [0.5,1.5], and [1.5,2.5] and we find the roots are at -0.7209, -0.3310, 1.144, 2.116, which gives us our θ which we use to find our four poses. When we enter each θ into the equations given to us in our problem description, we are returned the four poses which correspond respectively to the roots, -0.7209, -0.3310, 1.144, 2.116:

We also find that for each pose, the strut lengths match the given information. Next, we change the strut length p₂ to 7.00000000000000001, we then run our modified MATLAB script, “part5”, and we see that the graph of our new function looks like:

which we inspect and use the intervals [-1, -0.5], [-0.5, -0.3], [0, 0.1], [0.4, 0.5], [0.8, 1], [2, 3] to give to our bisection script. We are then returned the six roots: -0.6729, -0.3555, 0.0383, 0.4586, 0.9773, 2.515. We run this through our plotting algorithm in the our “part5” script and we are given the six poses:

We also find that for each pose, the strut lengths match the given information. After that, we set the strut length p₂=4. Running this through our script “part6”, we are presented with the graph:

which we analyze and give our bisection method the intervals [1.3, 1.4] and [1.7, 1.8]. We then find that our roots are 1.332 and 1.777 which we use to plot the poses and are returned the figures:

We also find that for each pose, the strut lengths match the given information.

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