Planar 1 DOF mechanism design is truly one of my favorite topics, so please forgive the long winded post, but it is well worth the read.
Consider this a "crash course" in mechanisms, which by itself could be an entire 4 year degree, but I will try to be brief.
There are two ways that you could go about the synthesis of this mechanism (there are actually more ways, but these are the two that you SHOULD consider)
1. Utilize complex loop closure equations to develop a mathematical model for finite position synthesis (by complex, I mean complex numbers, not computationally complex)
2. Use GCP techniques (I'll come back to this later) in CAD software to build a virtual model of the mechanism's motion
Number two is very very fast if the user is familiar with mechanisms and the CAD software, however it offers no information as to what the link velocities, or accelerations may be, meaning that you will not be able to determine the loads that the mechanism is under during motion. Option one can be relatively slow (that is unless you already have a code/calculation that you can modify), but it gives every bit of information required about the mechanism.
Since you would like to start sketching mechanisms immediately, it seems that option two is going to be your best bet. The acronym "GCP" refers to "graphical constraint programming." And, if you are familiar with loop closure equations and what kinematic constraints actually are, it is easy to understand what is going on in the background of the software. To describe GCP completely in this post would be quite difficult, so I will leave it up to you and google, but ASME has some excellent articles about this topic.
Before you begin with option two, you must first decide the mechanism "task", that is, what type of motion are you trying to accomplish with the mechanism? In this case, it would seem that you have two options:
1. Motion generation - in which both the path and the orientation of the body matters
2. Path generation - in which only the path of the object matters, and the orientation is of no concern
There are other tasks, but I don't think you will be requiring them for this specific application.
Many situations call for option 1, but you will have to make that choice yourself.
Once you have the task defined, you must determine the design positions - remember, there is a finite number of positions that we can design for the mechanism motion (in the case of motion generation, the theoretical maximum number of design positions is five, which is due to the number of free choices that one can make regarding the ground positions, drive link angles, etc. being zero, but I digress.) That is not to say that you cannot calculate what the path will be at discrete points using what is called position analysis, but we can only DESIGN for a specific number of locations.
In your case, you state "from A to B", which might initially lead you to think that you only need two design positions. However, I would warn you that in some cases, this may not be a good choice because your mechanism could behave erratically between your design positions, and the mechanism may be more likely to experience circuit or branch defects. In this case, I would aim for 3 position motion generation, it is usually a good place to start. This will limit some of your "free choices" but will have a higher probability of providing a robust design.
Now that you have an understanding of what task your mechanism needs to perform, it is time to select the appropriate linkage type, and inversion. This will depend on how complex your motion is, and a handful of other requirements such as packaging, required input torque, etc. However, in my experience, most problems can be solved with a standard 4-bar linkage (as opposed to six bar or eight bar linkages), which is quite easy to synthesize using GCP techniques. If you do choose a four-bar, you will have to determine if you want your drive link to be fully rotatable (for instance - if you wanted to drive it with an electric motor), or if you want it to have a limited travel range, in which case you would most likely need to drive it with an actuator (brief note: if the actuator is the linear type, then you would be creating a Watt II 6 bar chain!). Some good terms to search for are: "rocker-crank", "crank-rocker", "double-crank", "double-rocker", and Grashof condition.
Here is a good example of a 4-bar mechanism that I designed (in MATLAB) that performs 3 position motion generation, with a fully rotatable input link:
Hopefully that helps to give a little bit of context of what was previously discussed. The red circles are the design positions, and the link passing through them is called the coupler link, which is the link that provides the most complex motion in a four bar mechanism, highly non-linear as you can see.
There will be some other items of concern, such as circuit defects, branch defects, toggle positions, poor transmission angles, the list goes on, but what I have written should at least point you in the right direction. Some good names to google are: Arthur Erdman, George Sandor, Thomas Chase, John Mirth, and Larry Powell (compliant mechanisms)
Should you need additional assistance, please don't hesitate to let me know. I hope this was helpful, and wow, what a first post!
On a side note, should you start having to design a lot of mechanisms, building a mathematical model is the superior method, as it will allow you to determine every bit of knowledge that you want about the mechanism. Position, velocity, acceleration (which becomes very very important in fast moving mechanisms), input torque, mechanical advantage etc. You can even start getting fancy and OPTIMIZING these mechanisms!
