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Solve Equations Symbolically & Numerically

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Introduction

 

These are the position constraints from the multibody analysis of a double pendulum

 

posConscosθ__1Tx+sinθ__2+sinθ__1Tysinθ__1Tx1cosθ__2+cosθ__1Ty:

 

Symbolic Solution

 

We now symbolically solve the equations for the joint angles with solve

sol  solveposCons1, posCons2, θ__1,θ__2,explicit;

solθ__1=arctanTx2Ty2+Tx2Ty+Ty3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty2TyTx2+Ty2Tx,Tx2Ty+Ty3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty2Tx2+Ty2,θ__2=arctan2Tx22Ty2Tx2Ty+Ty3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty24TxTx2+Ty2+Tx2Ty+Ty32Tx,Tx22+Ty221,θ__1=arctanTx2Ty2Tx2TyTy3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty2TyTx2+Ty22Tx,Tx2TyTy3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty22Tx2+Ty2,θ__2=arctan2Tx22Ty2Tx2TyTy3+Tx62Tx4Ty2Tx2Ty4+4Tx4+4Tx2Ty24TxTx2+Ty2+Tx2Ty+Ty32Tx,Tx22+Ty221

(1)

Numeric Solution

 

We now numerically solve the equations for the joint angles with fsolve.

 

posCons2evalposCons,Tx=0.5,Ty=0.5

posCons20.5cosθ__1+sinθ__2+0.5sinθ__10.5sinθ__11cosθ__2+0.5cosθ__1

(2)

theta3_solfsolveposCons21,posCons22,θ__1=0..2Pi,θ__2=0..2Pi

theta3_solθ__1=4.288357941,θ__2=2.418858406

(3)

 

 

Applications

PV Diode Parameter Estimation

Flow through an Expansion Valve

Three Reservoir Problem