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I think this paper will help you. Analytical Inverse Kinematics and Self-motion Application for 7-DOF Redundant Manipulator.


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In general, when teaching my students how to do IK my first recommendation is always to find a variable (length/angle/value/etc) that is only a function of one joint so that they have one equation with one unknown. They can then look for another variable with is a function of one joint, or that is a function of two joints where one is the newly-solved-for ...


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I don't have the time to think this all the way through, but my gut reaction here is that you don't really have two degrees of freedom with those two joints because they're actuated together. It's not clear to me what exactly you mean when you say, "q2 and q3 is actuated together by one hydraulic piston" - is there literally just one piston? If so ...


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I believe it is standard to use "Matrix Algebra" i.e. matrix representation for solving kinematics-related problems (more generally, solving system of equations) rather than "Trigonometry". Matrix representation is compact and more efficient computationally speaking.


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