You need to solve it iteratively by using the pseudo-inverse of the transformation matrix of the direct kinematics.

Hey there Amin,

I am assuming that the manipulator looks something like this from the top, maybe with one additional joint.

http://www.intechopen.com/source/html/15855/media/image7.png

This is "redundant" manipulator, otherwise known as an over-actuated manipulator, and there is no absolute solution for IK for something like this that is over-actuated. That is to say, there are an infinite number of algebraic solutions. This is one of the reasons snake robots (a classic example of a manipulator being over-actuated in a certain plane) are harder to control ( I am again assuming you want the IK to control the manipulator). There are 3 things you can do here:

1) over specify you solution, which is to say, control for more than just the end effector. in this case you would have a second angle beta, which might be the orientation at the end of an inner link, and solve for that explicitly as your fourth equation. You will still have "elbow-out, elbow-in" decisions to specify at this point, but you will have a deterministic maximum on the number of solutions.

2) you can restrict some joints, either in control law or intrinsically. By adding hard join limits, or telling a joint to be fixed in a single position unless an event occurs, you have effectively turned your robot into a 3 DOF robot. This happens often in industry to avoid joint singularities: an extra join is strategically added and then kept fixed until a singularity occurs, which is restricting *that* joint, so adding in the "backup" joint can often pull you out of a singularity (this is also why the ISS and other large spacecraft use arrays of 4 Control Moment Gyroscopes instead of 3, one for every axis of rotation).

3) an iterative solution method. This is the most popular because usually manipulators are overactuated because their purpose is to be dynamically more free, but not so specific in purpose so as to be able to restrict them using methods (1) and (2). Bio-inspired bipedal legs like on the ATLAS robot are a good example of this http://cdni.wired.co.uk/620x413/a_c/bostonatlas.jpg

The iterative solution you implement is a little dependent on what you end up programming in to control the robot, but the gist is the same: each join is adjusted in a small amount very quickly in a way that moves the end effector toward the point and orientation of interest. You will face dramatically more extensive "elbow-out, elbow-in" problems here, as the algorithm will be trying to optimize the joint values amongst an infinite number of solutions, and so what most people do in this case is set an optimal or starting value for each joint, and then a weight that drives the joint toward that value. This way, the algorithm can optimize the weights and put your joints in an optimal configuration. Check out these links:

http://www.cescg.org/CESCG-2002/LBarinka/paper.pdf

http://www.mathworks.com/matlabcentral/fileexchange/29369-examples-of-basic-iterative-algorithms-for-inverse-kinematics

http://andreasaristidou.com/publications/FABRIK.pdf

http://sourceforge.net/apps/mediawiki/roblib/index.php?title=Robotics_Library

http://java.dzone.com/articles/tutorial-%E2%80%93-javafx-library

Hope this helps! Sorry its so long

Dear Amin,

You can use the following of the following approaches in association with planar robot,

1. Geometrical approach which is easy to deal with it in relation to planar robot

2. Direct method that link between the forward kinematic of the robot and the symbolic information of the required position and orientation. Accordingly try to find relation between both side of the equation to formulate the relation of the unknowns.

3. Iterative approach to resolve coupling between the unknowns. This is done by moving one of the matrices making the forward kinematics at a time to the symbolic matrix side, and then for each step apply the direct method.

Good luck

I suggest you to read Gosselin's articles, they will give you ideas, e.g.:

http://mechanicaldesign.asmedigitalcollection.asme.org/article.aspx?articleid=1453037