How can I obtain a unique solution for my index finger modeling problem?

[realriver]

I am trying to build an index finger model but come to a problem, which I can not get over, on abduction/adduction movement at metacarpophalangeal(MCP) joint.

Model description:
My model is used to predict fingertip isometric force generation. There are 3 joints considered. There is two DoF at MCP joint, which are flexion/extention and abduction/adduction. When I use my model (in Simulink) to predict isometric force at fingertip at different finger postures (finger may be bended or extended), I fix the fingertip at its place. To simulate abduction/adduction force, one muscle pulling force vector generate forces on the finger.

Problem description:
When finger is at a bended posture, fixed at the fingertip. Prediction of fingertip force became an underdetermined problem. To think about this, you can simply ignore the two distal joints and the flexion/extension Dof at MCP. Because they don't contribute to abduction/adduction at all. So the system became one body with two sides constrained. One terminal is fixed solidly (fingertip), one terminal is connected with a revolute joint (abd/add DoF). Then I can get infinite number of solutions for fingertip force generate, if I consider the finger is 'rigid'. Two typical solutions are as following. One is that the fingertip fixing point exert a force (but no torque) on fingertip to balance the muscle pulling force, but if the the fixing point is very close to the abd/add axis, the force goes to be huge (more than 10000N, which is impossible). The second one is that the fingertip fixing point exert only torque (no force, but MCP joint connection will take the force to balance muscle force vector) , this is more likely the case actually happening. But actually there are numbers of possible solutions between these two.

So, how to get a unique solution of the problem? What else I should consider? Like compliance of the finger?

 

[Unrest]

One is that the fingertip fixing point exert a force (but no torque) on fingertip to balance the muscle pulling force, but if the the fixing point is very close to the abd/add axis, the force goes to be huge (more than 10000N, which is impossible).

The second one is that the fingertip fixing point exert only torque (no force, but MCP joint connection will take the force to balance muscle force vector) , this is more likely the case actually happening. But actually there are numbers of possible solutions between these two.

I'm not really clear on your problem. Does it only occur when the finger tip is on or near the abd/add axis? Yes, you should expect the force to approach infinity because of the enormous mechanical advantage. In real life, the joints would deform to prevent such a large force appearing.

 

[nvn]

realriver: I currently think the constraint at the fingertip should be a ball joint. I.e., it should not resist torque. I currently think you should not be getting infinite force. You might be making a mistake in your free-body diagram. But we need more information, to be able to answer your question.

First, you need to draw a proper free-body diagram. Your current diagram is a good pictorial view, but it does not define the problem you are asking about. A free-body diagram contains labeled points, dimensions, and it contains coordinates and directions of applied forces and reaction forces. If it is a two-dimensional problem, it requires one diagram view. If it is a three-dimensional problem, it requires at least two orthographic views; or it requires one isometric view with (x, y, z) coordinates of each point labeled, and with the (x, y, z) unit vectors of applied forces and reaction forces labeled. Can you provide these coordinates, axis directions, and force directions?

You can label points on your diagram A, B, C, etc. Then list the (x, y, z) coordinates, and unit vector directions, in your post text.

We can speculate regarding the direction of your abduction/adduction axis and the direction of your applied force, but I cannot be sure of these directions from your current diagram. I think your applied force might (?) be drawn wrong, but I cannot determine what plane it is in, from your current diagram.