I have to solve a system of kinematic equations for a gripper mechanism. I tried to solve this with matrices but unfortunately there are too few independent equations to substitute trygonometric functions with variables. I need an analytital solution so numerical methods are insufficient. I also did a simulation model of the mechanism and the results seem to be linear, nevertheless the nonlinearities might just not be visible with bare eye.
Unknown: $\alpha2, \alpha3, \alpha45, \alpha10$
To derive: $\alpha3(v1), \alpha45(v1)$
$v2\cos(\alpha2)+v3\cos(\alpha3)+(v4+v5)\cos(\alpha45)+v6\cos(\alpha45+\alpha6)=-v1$ $v2\sin(\alpha2)+v3\sin(\alpha3)+(v4+v5)\sin(\alpha45)+v6\sin(\alpha45+\alpha6)=v7$ $v2\cos(\alpha2+v5\cos(\alpha45)+v6\cos(\alpha45+\alpha6)+v8\cos(\alpha2+\alpha8)+v10\cos(\alpha10)=-v1$ $v2\sin(\alpha2+v5\sin(\alpha45)+v6\sin(\alpha45+\alpha6)+v8\sin(\alpha2+\alpha8)+v10\sin(\alpha10)=v7$