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Wasn't sure whether this fits best in the maths or engineering forums, either way I'm hoping that a mathematical solution exists for my problem which I'll now explain.

My problem is one of biomechanics. One of the primary exercises I use is the barbell squat. Holding a loaded bar on your upper back and squatting down means that the skeleton of your body moves, the joints move and the bones and muscles transmit force through them down to the floor. I am looking for a mathematical model which can take the various measurements of the body and calculate the angles the joints are making with each other, the forces that are being transmitted through each member and the moments that exists around the joints.

Here is an image showing the position of the skeleton in the squat.

The load is positioned on the upper back, and L1 represents the length from the load to the hip joint. L2 is the length of the upper leg. The lower leg of length L3 pivots at the ankle and L4 represents the height of the heel on the shoe, which serves to lift the heel of the foot off the floor slightly. L5 is the length of the foot and the load must always remain on the centreline of the foot, as represented by the red centreline. The load always moves vertically on this centreline. The only inputs to the calculation are the lengths L1 to L5, and the size of the load in kilograms.

A1 is the angle formed between the spine and the upper leg at the hip. A2 is the angle formed between the upper leg and the lower leg at the knee. A3 is the angle formed between the lower leg and the foot at the ankle. What are these angles for any lengths of L1 to L5?

The load is stationary in this position, so the system must be in equilibrium. The load must be being transmitted to the ground through the bones. Given the size of the load, the various lengths L1 to L5, and the angles they are creating, what are the forces in each segment and what are the moments being created around each joint?

Im looking for a set of equations that can describe this model which I can plug in various lengths of L1 to L5 and a size of load variable.

Many thanks and looking forward to any responses!

Dan

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# Mathematical solution needed for this biomechanics problem

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